Annals of Communications in Mathematics: Volume 1, Number 1 (2018)

Madhumangal Pal; Austine Paul; Loganathan Karuppusamy; G. Muhiuddin

ISSN: 2582-0818 Volume 1, Number 1 (2018) An International Journal http://www.technoskypub.com/journal/acm/ Annals of Communications in Mathematics Volume 1, Number 1 (2018) Table of Contents Articles 1. Neutrosophic subsemigroups G. Muhiuddin Annals of Communications in Mathematics, Vol. 1 (1) (2018), 1-10 2. Hybrid structures and applications Y. B. Jun, S. Z. Song and G. Muhiuddin Annals of Communications in Mathematics, Vol. 1 (1) (2018), 11-25 3. p-semisimple neutrosophic quadruple BCI-algebras and neutrosophic quadruple p-ideals G. Muhiuddin and Y. B. Jun Annals of Communications in Mathematics, Vol. 1 (1) (2018), 26-37 4. An object oriented approach to the application of intuitionistic fuzzy sets in competency based test evaluation P. A. Ejegwa and I. C. Onyeke Annals of Communications in Mathematics, Vol. 1 (1) (2018), 38-47 5. Fuzzy shortest path in an interval-valued fuzzy hypergraph using similarity measures T. Pramanik and M. Pal Annals of Communications in Mathematics, Vol. 1 (1) (2018), 48-64 6. Cubic subalgebras of BCH-algebras T. Senapati and K.P. Shum Annals of Communications in Mathematics, Vol. 1 (1) (2018), 65-73 7. Generalized symmetric bi-derivations of lattices C. Jana and M. Pal Annals of Communications in Mathematics, Vol. 1 (1) (2018), 74-84 ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 1, Number 1 (2018), 1-10 ISSN: 2582-0818 c http://www.technoskypub.com NEUTROSOPHIC SUBSEMIGROUPS G. MUHIUDDIN A BSTRACT. In the present paper, we introduce the notion of (Φ, Ψ)-neutrosophic subsemigroups of a semigroup where Φ, Ψ ∈ {∈, q, ∈ ∨ q}, and related properties are investigated. We consider characterizations of an (∈, ∈)-neutrosophic subsemigroup and an (∈, ∈ ∨ q)-neutrosophic subsemigroup. Conditions for the neutrosophic ∈-subsets, neutrosophic q-subsets and neutrosophic ∈ ∨ q-subsets to be subsemigroups are discussed. Finally, we discuss conditions for a neutrosophic set to be a (q, ∈ ∨ q)-neutrosophic subsemigroup. 1. I NTRODUCTION The notion of neutrosophic set (NS) developed by Smarandache [8, 9, 10] is a more general platform which extends the concepts of the classic set and fuzzy set, intuitionistic fuzzy set and interval valued intuitionistic fuzzy set. Nowadays, the theory of neutrosophic sets became a very interesting and challenging topic of study as research point of view. The theory of neutrosophic sets is an important mathematical tool to deal with the indeterminate information and inconsistent information; and has vast applications in various directions (see e.g., [3], [4],[11], [11]). For more details we refer readers to the site http://fs.gallup.unm.edu/neutrosophy.htm. In the year 2015, Agboola and Davvaz introduced the concept of neutrosophic BCI/BCK-algebras and presented elementary properties of neutrosophic BCI/BCK-algebras. Further in the same year, they studied neutrosophic ideals of neutrosophic BCI-algebras (see [1] [2]). Recently, Muhiuddin et al. studied the notion of (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras [11]. Motivated by a lot of work on neutrosophic sets in various fields of research, in this paper, we introduce the notion of (Φ, Ψ)-neutrosophic subsemigroup of a semigroup S for Φ, Ψ ∈ {∈, q, ∈ ∨ q}, and investigate related properties. We provide characterizations of an (∈, ∈)-neutrosophic subsubsemigroup and an (∈, ∈ ∨ q)-neutrosophic subsubsemigroup. Given special sets, so called neutrosophic ∈-subsets, neutrosophic q-subsets and neutrosophic ∈ ∨ q-subsets, we provide conditions for the neutrosophic ∈-subsets, neutrosophic 2010 Mathematics Subject Classification. 06F35, 03G25, 03B52. Key words and phrases. (∈, ∈)-neutrosophic subsemigroup, (∈, q)-neutrosophic subsemigroup, (q, ∈)neutrosophic subsemigroup, (q, q)-neutrosophic subsemigroup, (∈, ∈ ∨ q)-neutrosophic subsemigroup, (q, ∈ ∨ q)-neutrosophic subsemigroup. 1 2 G. MUHIUDDIN q-subsets and neutrosophic ∈ ∨ q-subsets to be subsemigroups. We discuss conditions for a neutrosophic set to be a (q, ∈ ∨ q)-neutrosophic subsubsemigroup. 2. P RELIMINARIES Let S be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows: AB = {ab ∈ S | a ∈ A and b ∈ B} . Let S be a semigroup. By a subsemigroup of S we mean a nonempty subset A of S such that A2 ⊆ A. A fuzzy set A in a set S of the form t ∈ (0, 1] if y = x, A(y) := (2.1) 0 if y 6= x, is said to be a fuzzy point with support x and value t and is denoted by (x, t). For a fuzzy set A in a set S, a fuzzy point (x, t) is said to • contained in A, denoted by (x, t) ∈ A (see [7]), if A(x) ≥ t. • be quasi-coincident with A, denoted by (x, t) q A (see [7]), if A(x) + t > 1. For a fuzzy point (x, t) and a fuzzy set A in a set S, we say that • (x, t) ∈ ∨ q A if (x, t) ∈ A or (x, t) q A. For any family {ai | i ∈ Λ} of real numbers, we define _ max{ai | i ∈ Λ} if Λ is finite, {ai | i ∈ Λ} := sup{ai | i ∈ Λ} otherwise. ^ min{ai | i ∈ Λ} if Λ is finite, {ai | i ∈ Λ} := inf{ai | i ∈ Λ} otherwise. W If V Λ = {1, 2}, we will also use a1 ∨ a2 and a1 ∧ a2 instead of {ai | i ∈ Λ} and {ai | i ∈ Λ}, respectively. Let S be a non-empty set. A neutrosophic set (NS) in S (see [9]) is a structure of the form: A := {hx; AT (x), AI (x), AF (x)i | x ∈ S} where AT : S → [0, 1] is a truth membership function, AI : S → [0, 1] is an indeterminate membership function, and AF : S → [0, 1] is a false membership function. For the sake of simplicity, we shall use the symbol A = (AT , AI , AF ) for the neutrosophic set A := {hx; AT (x), AI (x), AF (x)i | x ∈ S}. NEUTROSOPHIC SUBSEMIGROUPS 3 3. N EUTROSOPHIC SUBSEMIGROUPS OF SEVERAL TYPES Given a neutrosophic set A = (AT , AI , AF ) in a set S, α, β ∈ (0, 1] and γ ∈ [0, 1), we consider the following sets: T∈ (A; α) := {x ∈ S | AT (x) ≥ α}, I∈ (A; β) := {x ∈ S | AI (x) ≥ β}, F∈ (A; γ) := {x ∈ S | AF (x) ≤ γ}, Tq (A; α) := {x ∈ S | AT (x) + α > 1}, Iq (A; β) := {x ∈ S | AI (x) + β > 1}, Fq (A; γ) := {x ∈ S | AF (x) + γ < 1}, T∈∨ q (A; α) := {x ∈ S | AT (x) ≥ α or AT (x) + α > 1}, I∈∨ q (A; β) := {x ∈ S | AI (x) ≥ β or AI (x) + β > 1}, F∈∨ q (A; γ) := {x ∈ S | AF (x) ≤ γ or AF (x) + γ < 1}. We say T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are neutrosophic ∈-subsets; Tq (A; α), Iq (A; β) and Fq (A; γ) are neutrosophic q-subsets; and T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are neutrosophic ∈ ∨ q-subsets. For Φ ∈ {∈, q, ∈ ∨ q}, the element of TΦ (A; α) (resp., IΦ (A; β) and FΦ (A; γ)) is called a neutrosophic TΦ -point (resp., neutrosophic IΦ -point and neutrosophic FΦ -point) with value α (resp., β and γ). It is clear that T∈∨ q (A; α) = T∈ (A; α) ∪ Tq (A; α), (3.1) I∈∨ q (A; β) = I∈ (A; β) ∪ Iq (A; β), (3.2) F∈∨ q (A; γ) = F∈ (A; γ) ∪ Fq (A; γ). (3.3) Definition 3.1. Given Φ, Ψ ∈ {∈, q, ∈ ∨ q}, a neutrosophic set A = (AT , AI , AF ) in a semigroup S is called a (Φ, Ψ)-neutrosophic subsemigroup of S if the following assertions are valid. x ∈ TΦ (A; αx ), y ∈ TΦ (A; αy ) ⇒ xy ∈ TΨ (A; αx ∧ αy ), x ∈ IΦ (A; βx ), y ∈ IΦ (A; βy ) ⇒ xy ∈ IΨ (A; βx ∧ βy ), (3.4) x ∈ FΦ (A; γx ), y ∈ FΦ (A; γy ) ⇒ xy ∈ FΨ (A; γx ∨ γy ) for all x, y ∈ S, αx , αy , βx , βy , ∈ (0, 1] and γx , γy ∈ [0, 1). Theorem 3.1. A neutrosophic set A = (AT , AI , AF ) in a semigroup S is an (∈, ∈)neutrosophic subsemigroup of S if and only if it satisfies:   AT (xy) ≥ AT (x) ∧ AT (y)   (∀x, y ∈ S)  AI (xy) ≥ AI (x) ∧ AI (y)  . (3.5) AF (xy) ≤ AF (x) ∨ AF (y) Proof. Assume that A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic subsemigroup of S. If there exist x, y ∈ S such that AT (xy) < AT (x) ∧ AT (y), then AT (xy) < αt ≤ AT (x) ∧ AT (y) for some αt ∈ (0, 1]. It follows that x, y ∈ T∈ (A; αt ) but xy ∈ / T∈ (A; αt ). Hence AT (xy) ≥ AT (x) ∧ AT (y) for all x, y ∈ S. Similarly, we show that AI (xy) ≥ AI (x) ∧ AI (y) 4 G. MUHIUDDIN for all x, y ∈ S. Suppose that there exist a, b ∈ S and γf ∈ [0, 1] be such that AF (ab) > γf ≥ AF (a) ∨ AF (b). Then a, b ∈ F∈ (A; γf ) and ab ∈ / F∈ (A; γf ), which is a contradiction. Therefore AF (xy) ≤ AF (x) ∨ AF (y) for all x, y ∈ S. Conversely, let A = (AT , AI , AF ) be a neutrosophic set in S which satisfies the condition (3.5). Let x, y ∈ S be such that x ∈ T∈ (A; αx ) and y ∈ T∈ (A; αy ). Then AT (x) ≥ αx and AT (y) ≥ αy , which imply that AT (xy) ≥ AT (x) ∧ AT (y) ≥ αx ∧ αy , that is, xy ∈ T∈ (A; αx ∧ αy ). Similarly, if x ∈ I∈ (A; βx ) and y ∈ I∈ (A; βy ) then xy ∈ I∈ (A; βx ∧ βy ). Now, let x ∈ F∈ (A; γx ) and y ∈ F∈ (A; γy ) for x, y ∈ S. Then AF (x) ≤ γx and AF (y) ≤ γy , and so AF (xy) ≤ AF (x) ∨ AF (y) ≤ γx ∨ γy . Hence xy ∈ F∈ (A; γx ∨ γy ). Therefore A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic subsemigroup of S. Theorem 3.2. If A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic subsemigroup of a semigroup S, then neutrosophic q-subsets Tq (A; α), Iq (A; β) and Fq (A; γ) are subsemigroups of S for all α, β ∈ (0, 1] and γ ∈ [0, 1) whenever they are nonempty. Proof. Let x, y ∈ Tq (A; α). Then AT (x) + α > 1 and AT (y) + α > 1. It follows that AT (xy) + α ≥ (AT (x) ∧ AT (y)) + α = (AT (x) + α) ∧ (AT (y) + α) > 1 and so that xy ∈ Tq (A; α). Hence Tq (A; α) is a subsemigroup of S. Similarly, we can prove that Iq (A; β) is a subsemigroup of S. Now let x, y ∈ Fq (A; γ). Then AF (x)+γ < 1 and AF (y) + γ < 1, which imply that AF (xy) + γ ≤ (AF (x) ∨ AF (y)) + γ = (AF (x) + α) ∨ (AF (y) + α) < 1. Hence xy ∈ Fq (A; γ) and Fq (A; γ) is a subsemigroup of S. Theorem 3.3. If A = (AT , AI , AF ) is a (q, ∈ ∨ q)-neutrosophic subsemigroup of a semigroup S, then neutrosophic q-subsets Tq (A; α), Iq (A; β) and Fq (A; γ) are subsemigroups of S for all α, β ∈ (0.5, 1] and γ ∈ [0, 0, 5) whenever they are nonempty. Proof. Let x, y ∈ Tq (A; α). Then xy ∈ T∈∨ q (A; α), and so xy ∈ T∈ (A; α) or xy ∈ Tq (A; α). If xy ∈ T∈ (A; α), then AT (xy) ≥ α > 1 − α since α > 0.5. Hence xy ∈ Tq (A; α). Therefore Tq (A; α) is a subsemigroup of S. Similarly, we prove that Iq (A; β) is a subsemigroup of S. Let x, y ∈ Fq (A; γ). Then xy ∈ F∈∨ q (A; γ), and so xy ∈ F∈ (A; γ) or xy ∈ Fq (A; γ). If xy ∈ F∈ (A; γ), then AF (xy) ≤ γ < 1 − γ since γ ∈ [0, 0, 5). Hence xy ∈ Fq (A; γ), and therefore Fq (A; γ) is a subsemigroup of S. We provide characterizations of an (∈, ∈ ∨ q)-neutrosophic subsemigroup. Theorem 3.4. A neutrosophic set A = (AT , AI , AF ) in a semigroup S is an (∈, ∈ ∨ q)neutrosophic subsemigroup of S if and only if it satisfies:   V AT (xy) ≥ {AT (x), AT (y), 0.5} V   (∀x, y ∈ S)  AI (xy) ≥ {AI (x), AI (y).0.5}  . (3.6) W AF (xy) ≤ {AF (x), AF (y), 0.5} Proof. Suppose that A = (AT , AI , AF ) is an (∈, ∈ ∨ q)-neutrosophic subsemigroup of S and let x, y ∈ S. If AT (x) ∧ AT (y) < 0.5, then AT (xy) ≥ AT (x) ∧ AT (y). For, assume that AT (xy) < AT (x) ∧ AT (y) and choose αt such that AT (xy) < αt < AT (x) ∧ AT (y). NEUTROSOPHIC SUBSEMIGROUPS 5 Then x ∈ T∈ (A; αt ) and y ∈ T∈ (A; αt ) but xy ∈ / T∈ (A; αt ). Also AT (xy) + αt < 1, i.e., xy ∈ / T (A; α ). Thus xy ∈ / T (A; α ), a contradiction. Therefore AT (xy) ≥ q t ∈∨ q t V {AT (x), AT (y), 0.5} whenever AT (x) ∧ AT (y) < 0.5. Now suppose that AT (x) ∧ AT (y) ≥ 0.5. Then x ∈ T∈ (A; 0.5) and y ∈ T∈ (A; 0.5), which imply that xy ∈ T∈∨ q (A; 0.5). Hence AT (xy) ≥ 0.5. Otherwise, AT (xy) + 0.5 < 0.5 + 0.5 = 1, a V contradiction. Consequently, AT (xy) ≥ {A (x), AT (y), 0.5} for all x, y ∈ S. SimT V ilarly, we know that AI (xy) ≥ {AI (x), AI (y), 0.5} for all x, y ∈ S. Suppose that AF (x) ∨ AF (y) > 0.5. If AF (xy) > AF (x) ∨ AF (y) := γf , then x, y ∈ F∈ (A; γf ), xy ∈ / F∈ (A; γf ) and AFW(xy) + γf > 2γf > 1, i.e., xy ∈ / Fq (A; γf ). This is a contradiction. Hence AF (xy) ≤ {AF (x), AF (y), 0.5} whenever AF (x)∨AF (y) > 0.5. Now, assume that AF (x)∨AF (y) ≤ 0.5. Then x, y ∈ F∈ (A; 0.5) and so xy ∈ F∈∨ q (A; 0.5). Thus AF (xy) ≤ 0.5 or AF (xy) + 0.5 < 1. If AF (xy) > 0.5, thenWAF (xy) + 0.5 > 0.5 + 0.5 = 1, a contradiction. Thus AF (xy) ≤ 0.5, and so AFW(xy) ≤ {AF (x), AF (y), 0.5} whenever AF (x)∨AF (y) ≤ 0.5. Therefore AF (xy) ≤ {AF (x), AF (y), 0.5} for all x, y ∈ S. Conversely, let A = (AT , AI , AF ) be a neutrosophic set in S which satisfies the condition (3.6). Let x, y ∈ S and αx , αy , βx , βy , γx , γy ∈ [0, 1]. If x ∈ T∈ (A; αx ) and y ∈ T∈ (A; αy ), then AT (x) ≥ αx and AT (y) ≥ αy . If AT (xy) < αx ∧ αy , then AT (x) ∧ AT (y) ≥ 0.5. Otherwise, we have ^ AT (xy) ≥ {AT (x), AT (y), 0.5} = AT (x) ∧ AT (y) ≥ αx ∧ αy , a contradiction. It follows that AT (xy) + αx ∧ αy > 2AT (xy) ≥ 2 ^ {AT (x), AT (y), 0.5} = 1 and so that xy ∈ Tq (A; αx ∧ αy ) ⊆ T∈∨ q (A; αx ∧ αy ). Similarly, if x ∈ I∈ (A; βx ) and y ∈ I∈ (A; βy ), then xy ∈ I∈∨ q (A; βx ∧ βy ). Now, let x ∈ F∈ (A; γx ) and y ∈ F∈ (A; γy ). Then AF (x) ≤ γx and AF (y) ≤ γy . If AF (xy) > γx ∨ γy , then AF (x) ∨ AF (y) ≤ 0.5 because if not, then _ AF (xy) ≤ {AF (x), AF (y), 0.5} ≤ AF (x) ∨ AF (y) ≤ γx ∨ γy , which is a contradiction. Hence AF (xy) + γx ∨ γy < 2AF (xy) ≤ 2 _ {AF (x), AF (y), 0.5} = 1, and so xy ∈ Fq (A; γx ∨ γy ) ⊆ F∈∨ q (A; γx ∨ γy ). Therefore A = (AT , AI , AF ) is an (∈, ∈ ∨ q)-neutrosophic subsemigroup of S. Theorem 3.5. If A = (AT , AI , AF ) is an (∈, ∈ ∨ q)-neutrosophic subsemigroup of a semigroup S, then neutrosophic q-subsets Tq (A; α), Iq (A; β) and Fq (A; γ) are subsemigroups of S for all α, β ∈ (0.5, 1] and γ ∈ [0, 0.5) whenever they are nonempty. Proof. Assume that Tq (A; α), Iq (A; β) and Fq (A; γ) are nonempty for all α, β ∈ (0.5, 1] and γ ∈ [0, 0.5). Let x, y ∈ Tq (A; α). Then AT (x) + α > 1 and AT (y) + α > 1. It follows from Theorem 3.4 that ^ AT (xy) + α ≥ {AT (x), AT (y), 0.5} + α ^ = {AT (x) + α, AT (y) + α, 0.5 + α} > 1, that is, xy ∈ Tq (A; α). Hence Tq (A; α) is a subsemigroup of S. By the similar way, we can induce that Iq (A; β) is a subsemigroup of S. Now, let x, y ∈ Fq (A; γ). Then 6 G. MUHIUDDIN AF (x) + γ < 1 and AF (y) + γ < 1. Using Theorem 3.4, we have _ AF (xy) + γ ≤ {AF (x), AF (y), 0.5} + γ _ = {AF (x) + γ, AF (y) + γ, 0.5 + γ} < 1, and so xy ∈ Fq (A; γ). Therefore Fq (A; γ) is a subsemigroup of S. Theorem 3.6. For a neutrosophic set A = (AT , AI , AF ) in a semigroup S, if the nonempty neutrosophic ∈ ∨ q-subsets T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are subsemigroups of S for all α, β ∈ (0, 1] and γ ∈ [0, 1), then A = (AT , AI , AF ) is an (∈, ∈ ∨ q)neutrosophic subsemigroup of S. Proof. Let T∈∨ q (A; α) be a subsemigroup of S and assume that ^ AT (xy) < {AT (x), AT (y), 0.5} for some x, y ∈ S. Then there exists α ∈ (0, 0.5] such that ^ AT (xy) < α ≤ {AT (x), AT (y), 0.5}. It follows that x, y ∈ T∈ (A; α) ⊆ T∈∨ q (A; α), and so that xy ∈ T∈∨ q (A; α). Hence AT (xy) ≥ α or AT (xy) + α > 1. This is a contradiction, and so ^ AT (xy) ≥ {AT (x), AT (y), 0.5} for all x, y ∈ S. Similarly, we show that ^ AI (xy) ≥ {AI (x), AI (y), 0.5} for all x, y ∈ S. Now let F∈∨ q (A; γ) be a subsemigroup of S and assume that _ AF (xy) > {AF (x), AF (y), 0.5} for some x, y ∈ S. Then AF (xy) > γ ≥ _ {AF (x), AF (y), 0.5}, (3.7) for some γ ∈ [0.5, 1), which implies that x, y ∈ F∈ (A; γ) ⊆ F∈∨ q (A; γ). Thus xy ∈ F∈∨ q (A; γ). From (3.7), we have xy ∈ / F∈ (A; γ) and AF (xy) + γ > 2γ ≥ 1, i.e., xy ∈ / Fq (A; γ). This is a contradiction, and hence _ AF (xy) ≤ {AF (x), AF (y), 0.5} for all x, y ∈ S. Using Theorem 3.4, we know that A = (AT , AI , AF ) is an (∈, ∈ ∨ q)neutrosophic subsemigroup of S. Theorem 3.7. If A = (AT , AI , AF ) is an (∈, ∈ ∨ q)-neutrosophic subsemigroup of a semigroup S, then nonempty neutrosophic ∈ ∨ q-subsets T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are subsemigroups of S for all α, β ∈ (0, 0.5] and γ ∈ [0.5, 1). Proof. Assume that T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are nonempty for all α, β ∈ (0, 0.5] and γ ∈ [0.5, 1). Let x, y ∈ I∈∨ q (A; β). Then x ∈ I∈ (A; β) or x ∈ Iq (A; β), and y ∈ I∈ (A; β) or y ∈ Iq (A; β). NEUTROSOPHIC SUBSEMIGROUPS 7 Hence we have the following four cases: (i) (ii) (iii) (iv) x ∈ I∈ (A; β) and y ∈ I∈ (A; β), x ∈ I∈ (A; β) and y ∈ Iq (A; β), x ∈ Iq (A; β) and y ∈ I∈ (A; β), x ∈ Iq (A; β) and y ∈ Iq (A; β). The first case implies that xy ∈ I∈∨ q (A; β). For the second case, y ∈ Iq (A; β) induces AI (y) > 1 − β ≥ β, that is, y ∈ I∈ (A; β). Thus xy ∈ I∈∨ q (A; β). Similarly, the third case implies xy ∈ I∈∨ q (A; β). The last case induces AI (x) > 1 − β ≥ β and AI (y) > 1 − β ≥ β, that is, x ∈ I∈ (A; β) and y ∈ I∈ (A; β). Hence xy ∈ I∈∨ q (A; β). Therefore I∈∨ q (A; β) is a subsemigroup of S for all β ∈ (0, 0.5]. By the similar way, we show that T∈∨ q (A; α) is a subsemigroup of S for all α ∈ (0, 0.5]. Let x, y ∈ F∈∨ q (A; γ). Then AF (x) ≤ γ or AF (x) + γ < 1, and AF (y) ≤ γ or AF (y) + γ < 1. If AF (x) ≤ γ and AF (y) ≤ γ, then _ _ AF (xy) ≤ {AF (x), AF (y), 0.5} ≤ {γ, 0.5} = γ by Theorem 3.4, and so xy ∈ F∈ (A; γ) ⊆ F∈∨ q (A; γ). If AF (x) ≤ γ and AF (y)+γ < 1, then _ _ AF (xy) ≤ {AF (x), AF (y), 0.5} ≤ {γ, 1 − γ, 0.5} = γ by Theorem 3.4. Thus xy ∈ F∈ (A; γ) ⊆ F∈∨ q (A; γ). Similarly, if AF (x) + γ < 1 and AF (y) ≤ γ, then xy ∈ F∈∨ q (A; γ). Finally, assume that AF (x) + γ < 1 and AF (y) + γ < 1. Then _ _ AF (xy) ≤ {AF (x), AF (y), 0.5} ≤ {1 − γ, 0.5} = 0.5 < γ by Theorem 3.4. Hence xy ∈ F∈ (A; γ) ⊆ F∈∨ q (A; γ). Consequently, F∈∨ q (A; γ) is a subsemigroup of S for all γ ∈ [0.5, 1). Theorem 3.8. If A = (AT , AI , AF ) is a (q, ∈ ∨ q)-neutrosophic subsemigroup of a semigroup S, then nonempty neutrosophic ∈ ∨ q-subsets T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are subsemigroups of S for all α, β ∈ (0.5, 1] and γ ∈ [0, 0.5). Proof. Assume that T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are nonempty for all α, β ∈ (0.5, 1] and γ ∈ [0, 0.5). Let x, y ∈ T∈∨ q (A; α). Then x ∈ T∈ (A; α) or x ∈ Tq (A; α). and y ∈ T∈ (A; α) or y ∈ Tq (A; α). If x ∈ Tq (A; α) and y ∈ Tq (A; α), then obviously xy ∈ T∈∨ q (A; α). Suppose that x ∈ T∈ (A; α) and y ∈ Tq (A; α). Then AT (x) + α ≥ 2α > 1, i.e., x ∈ Tq (A; α). It follows that xy ∈ T∈∨ q (A; α). Similarly, if x ∈ Tq (A; α) and y ∈ T∈ (A; α), then xy ∈ T∈∨ q (A; α). Now, let x, y ∈ F∈∨ q (A; γ). Then x ∈ F∈ (A; γ) or x ∈ Fq (A; γ), 8 G. MUHIUDDIN and y ∈ F∈ (A; γ) or y ∈ Fq (A; γ). If x ∈ Fq (A; γ) and y ∈ Fq (A; γ), then clearly xy ∈ F∈∨ q (A; γ). If x ∈ F∈ (A; γ) and y ∈ Fq (A; γ), then AF (x) + γ ≤ 2γ < 1, i.e., x ∈ Fq (A; γ). It follows that xy ∈ F∈∨ q (A; γ). Similarly, if x ∈ Fq (A; γ) and y ∈ F∈ (A; γ), then xy ∈ F∈∨ q (A; γ). Finally, assume that x ∈ F∈ (A; γ) and y ∈ F∈ (A; γ). Then AF (x) + γ ≤ 2γ < 1 and AF (y) + γ ≤ 2γ < 1, that is, x ∈ Fq (A; γ) and y ∈ Fq (A; γ). Therefore xy ∈ F∈∨ q (A; γ). Consequently, T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are subsemigroups of S for all α, β ∈ (0.5, 1] and γ ∈ [0, 0.5). Given a neutrosophic set A = (AT , AI , AF ) in a set S, we consider: S01 := {x ∈ S | AT (x) > 0, AI (x) > 0, AF (x) < 1}. Theorem 3.9. If a neutrosophic set A = (AT , AI , AF ) in a semigroup S is an (∈, ∈)neutrosophic subsemigroup of S, then the set S01 is a subsemigroup of S. Proof. Let x, y ∈ S01 . Then AT (x) > 0, AI (x) > 0, AF (x) < 1, AT (y) > 0, AI (y) > 0 and AF (y) < 1. Suppose that AT (xy) = 0. Note that x ∈ T∈ (A; AT (x)) and y ∈ T∈ (A; AT (y)). But xy ∈ / T∈ (A; AT (x) ∧ AT (y)) because AT (xy) = 0 < AT (x) ∧ AT (y). This is a contradiction, and thus AT (xy) > 0. By the similar way, we show that AI (xy) > 0. Note that x ∈ F∈ (A; AF (x)) and y ∈ F∈ (A; AF (y)). If AF (xy) = 1, then AF (xy) = 1 > AF (x) ∨ AF (y), and so xy ∈ / F∈ (A; AF (x) ∨ AF (y)). This is impossible. Hence xy ∈ S01 , and therefore S01 is a subsemigroup of S. Theorem 3.10. If a neutrosophic set A = (AT , AI , AF ) in a semigroup S is an (∈, q)neutrosophic subsemigroup of S, then the set S01 is a subsemigroup of S. Proof. Let x, y ∈ S01 . Then AT (x) > 0, AI (x) > 0, AF (x) < 1, AT (y) > 0, AI (y) > 0 and AF (y) < 1. If AT (xy) = 0, then AT (xy) + AT (x) ∧ AT (y) = AT (x) ∧ AT (y) ≤ 1. Hence xy ∈ / Tq (A; AT (x) ∧ AT (y)), which is a contradiction since x ∈ T∈ (A; AT (x)) and y ∈ T∈ (A; AT (y)). Thus AT (xy) > 0. Similarly, we get AI (xy) > 0. Assume that AF (xy) = 1. Then AF (xy) + AF (x) ∨ AF (y) = 1 + AF (x) ∨ AF (y) ≥ 1, that is, xy ∈ / Fq (A; AF (x)∨AF (y)). This is a contradiction because of x ∈ F∈ (A; AF (x)) and y ∈ F∈ (A; AF (y)). Hence AF (xy) < 1. Consequently, xy ∈ S01 and S01 is a subsemigroup of S. Theorem 3.11. If a neutrosophic set A = (AT , AI , AF ) in a semigroup S is a (q, ∈)neutrosophic subsemigroup of S, then the set S01 is a subsemigroup of S. Proof. Let x, y ∈ S01 . Then AT (x) > 0, AI (x) > 0, AF (x) < 1, AT (y) > 0, AI (y) > 0 and AF (y) < 1. It follows that AT (x)+1 > 1, AT (y)+1 > 1, AI (x)+1 > 1, AI (y)+1 > 1, AF (x) + 0 < 1 and AF (y) + 0 < 1. Hence x, y ∈ Tq (A; 1) ∩ Iq (A; 1) ∩ Fq (A; 0). If AT (xy) = 0 or AI (xy) = 0, then AT (xy) < 1 = 1 ∧ 1 or AI (xy) < 1 = 1 ∧ 1. Thus xy ∈ / Tq (A; 1 ∧ 1) or xy ∈ / Iq (A; 1 ∧ 1), a contradiction. Hence AT (xy) > 0 and AI (xy) > 0. If AF (xy) = 1, then xy ∈ / Fq (A; 0 ∨ 0) which is a contradiction. Thus AF (xy) < 1. Therefore xy ∈ S01 and the proof is complete. NEUTROSOPHIC SUBSEMIGROUPS 9 Theorem 3.12. If a neutrosophic set A = (AT , AI , AF ) in a semigroup S is a (q, q)neutrosophic subsemigroup of S, then the set S01 is a subsemigroup of S. Proof. Let x, y ∈ S01 . Then AT (x) > 0, AI (x) > 0, AF (x) < 1, AT (y) > 0, AI (y) > 0 and AF (y) < 1. Hence AT (x) + 1 > 1, AT (y) + 1 > 1, AI (x) + 1 > 1, AI (y) + 1 > 1, AF (x) + 0 < 1 and AF (y) + 0 < 1. Hence x, y ∈ Tq (A; 1) ∩ Iq (A; 1) ∩ Fq (A; 0). If AT (xy) = 0 or AI (xy) = 0, then AT (xy) + 1 ∧ 1 = 0 + 1 = 1 or AI (xy) + 1 ∧ 1 = 0 + 1 = 1, and so xy ∈ / Tq (A; 1 ∧ 1) or xy ∈ / Iq (A; 1 ∧ 1). This is impossible, and thus AT (xy) > 0 and AI (xy) > 0. If AF (xy) = 1, then AF (xy)+0∨0 = 1, that is, xy ∈ / Fq (A; 0∨0). This is a contradiction, and so AF (xy) < 1. Therefore xy ∈ S01 and the proof is complete. We provide conditions for a neutrosophic set to be a (q, ∈ ∨ q)-neutrosophic subsemigroup. Theorem 3.13. For a subsemigroup Q of a semigroup S, let A = (AT , AI , AF ) be a neutrosophic set in S such that (∀x ∈ Q) (AT (x) ≥ 0.5, AI (x) ≥ 0.5, AF (x) ≤ 0.5) , (3.8) (∀x ∈ S \ Q) (AT (x) = 0, AI (x) = 0, AF (x) = 1) . (3.9) Then A = (AT , AI , AF ) is a (q, ∈ ∨ q)-neutrosophic subsemigroup of S. Proof. Assume that x ∈ Iq (A; βx ) and y ∈ Iq (A; βy ) for all x, y ∈ S and βx , βy ∈ [0, 1]. Then AI (x)+βx > 1 and AI (y)+βy > 1. If xy ∈ / Q, then x ∈ S \Q or y ∈ S \Q since Q is a subsemigroup of S. Hence AI (x) = 0 or AI (y) = 0, which imply that βx > 1 or βy > 1. This is a contradiction, and so xy ∈ Q. If βx ∧ βy > 0.5, then AI (xy) + βx ∧ βy > 1, i.e., xy ∈ Iq (A; βx ∧ βy ). If βx ∧ βy ≤ 0.5, then AI (xy) ≥ 0.5 ≥ βx ∧ βy , i.e., xy ∈ I∈ (A; βx ∧ βy ). Hence xy ∈ I∈∨ q (A; βx ∧ βy ). Similarly, if x ∈ Tq (A; αx ) and y ∈ Tq (A; αy ) for all x, y ∈ S and αx , αy ∈ [0, 1], then xy ∈ T∈∨ q (A; αx ∧ αy ). Now let x, y ∈ S and γx , γy ∈ [0, 1] be such that x ∈ Fq (A; γx ) and y ∈ Fq (A; γy ). Then AF (x)+γx < 1 and AF (y)+γy < 1. It follows that xy ∈ Q. In fact, if not then x ∈ S \Q or y ∈ S \ Q since Q is a subsemigroup of S. Hence AF (x) = 1 or AF (y) = 1, which imply that γx < 0 or γy < 0. This is a contradiction, and so xy ∈ Q. If γx ∨ γy ≥ 0.5, then AF (xy) ≤ 0.5 ≤ γx ∨ γy , that is, xy ∈ F∈ (A; γx ∨ γy ). If γx ∨ γy < 0.5, then AF (xy) + γx ∨ γy < 1, that is, xy ∈ Fq (A; γx ∨ γy ). Hence xy ∈ F∈∨ q (A; γx ∨ γy ), and consequently A = (AT , AI , AF ) is a (q, ∈ ∨ q)-neutrosophic subsemigroup of S. Combining Theorems 3.3 and 3.13, we have the following corollary. Corollary 3.14. For a subsemigroup Q of S, if A = (AT , AI , AF ) is a neutrosophic set in S satisfying conditions (3.8) and (3.9), then Tq (A; α), Iq (A; β) and Fq (A; γ) are subsemigroups of S for all α, β ∈ (0.5, 1] and γ ∈ [0, 0, 5) whenever they are nonempty. 4. C ONCLUSION In this paper, we introduce the notion of (Φ, Ψ)-neutrosophic subsemigroup of a semigroup S for Φ, Ψ ∈ {∈, q, ∈ ∨ q}, and investigate related properties. We provide characterizations of an (∈, ∈)-neutrosophic subsubsemigroup and an (∈, ∈ ∨ q)-neutrosophic 10 G. MUHIUDDIN subsubsemigroup. Given special sets, so called neutrosophic ∈-subsets, neutrosophic qsubsets and neutrosophic ∈ ∨ q-subsets, we provide conditions for the neutrosophic ∈subsets, neutrosophic q-subsets and neutrosophic ∈ ∨ q-subsets to be subsemigroups. We discuss conditions for a neutrosophic set to be a (q, ∈ ∨ q)-neutrosophic subsubsemigroup. We hope that this work will provide a deep impact on the upcoming research in this field and other related algebraic structures studies to open up new horizons of interest and innovations. As future directions, one can further study the neutrosophic set theory in different algebras such as BCK/BCI-algebras, BL-algebras, EQ-algebras, B-algebras, MV-algebras, Q-algebras, etc. 5. ACKNOWLEDGEMENTS The author is grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper. R EFERENCES [1] A.A.A. Agboola and B. Davvaz, On neutrosophic ideals of neutrosophic BCI-algebras, Critical Review. Volume X, (2015), 93–103. [2] A.A.A. Agboola and B. Davvaz, Introduction to neutrosophic BCI/BCK-algebras, Inter. J. Math. Math. Sci. Volume 2015, Article ID 370267, 6 pages. [3] S. Broumi and F. Smarandache, Correlation coefficient of interval neutrosophic sets. Appl. Mech. Mater. 436 (2013), 511517. [4] H.D. Cheng and Y. Guo, A new neutrosophic approach to image thresholding. New Math. Nat. Comput. 42(008), 291–308. [5] Y. Guo and H.D. Cheng, New neutrosophic approach to image segmentation. Pat. Recognit. 42 (2009), 587–595. [6] G. Muhiuddin, H. Bordbar, F. Smarandache, Y. B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras, Neutrosophic Sets and Systems, Vol. 20 (2018), 36–43. [7] P. M. Pu and Y. M. Liu, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980), 571–599. [8] F. Smarandache, Neutrosophy, Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, 105 p., 1998. http://fs.gallup.unm.edu/eBook-neutrosophics6.pdf (last edition online). [9] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Reserch Press, Rehoboth, NM, 1999. [10] F. Smarandache, Neutrosophic set-a generalization of the intuitionistic fuzzy set, Int. J. Pure Appl. Math. 24(3) (2005), 287–297. [11] J. Ye, Similarity measures between interval neutrosophic sets and their multicriteria decision-making method. J. Intell. Fuzzy Syst. 26 (2014), 165–172. G. M UHIUDDIN D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF TABUK , TABUK 71491, S AUDI A RABIA Email address: chishtygm@gmail.com ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 1, Number 1 (2018), 11-25 ISSN: 2582-0818 c http://www.technoskypub.com HYBRID STRUCTURES AND APPLICATIONS YOUNG BAE JUN, SEOK ZUN SONG AND G. MUHIUDDIN∗ A BSTRACT. The notion of hybrid structures is introduced, and several properties are investigated. Applications in BCK/BCI-algebras and linear spaces are discussed. Hybrid subalgebra, hybrid field and hybrid linear space are introduced, and related properties are investigated. 1. I NTRODUCTION The concept of fuzzy set is first introduced by Zadeh [21] in 1965. As a generalization of fuzzy sets, Torra introduced the concept of hesitant fuzzy sets ([19, 20]). The hesitant fuzzy set is very useful to express peoples hesitancy in daily life, and it is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. Various problems in many fields involve data containing uncertainties which are dealt with wide range of existing theories such as the theory of probability, (intuitionistic) fuzzy set theory, vague sets, theory of interval mathematics and rough set theory etc. All of these theories have their own difficulties which are pointed out in [15]. To overcome these difficulties, Molodtsov [15] introduced the soft set theory as a new mathematical tool for dealing with uncertainties that is free from the difficulties. Molodtsov successfully applied the soft set theory in several directions, such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, theory of measurement and so on (see [15, 16, 17, 18]). Soft set theory is applied to algebraic structures such as BCK/BCI-algebra ([8, 11, 12]), d-algebras ([10]), ring ([1]), semiring ([3, 6]), group (see [2]), ordered semigroup ([9]), decision making ([4, 5, 13]) and BL-algebra ([22]). As a parallel circuit of fuzzy sets and soft sets (or, hesitant fuzzy sets), we introduced the notion of hybrid structure in a set of parameters over an initial universe set, and investigate several properties. Using this notion, we introduce the concepts of a hybrid subalgebra, a hybrid field and a hybrid linear space. We consider the hybrid union and hybrid intersection of hybrid subalgebras in BCK/BCI-algebras. We discuss characterizations of a hybrid subalgebra and a hybrid linear space. Given a hybrid subalgebra, we make a new hybrid subalgebra in BCK/BCI-algebras. We consider the hybrid image and preimage of a 2010 Mathematics Subject Classification. 06D72, 06F35. Key words and phrases. Hybrid structure; hybrid subalgebra; hybrid field; hybrid linear space; hybrid union; hybrid intersection. *Corresponding author. 11 12 Y. B. JUN, S. Z. SONG AND G. MUHIUDDIN hybrid subalgebra and a hybrid linear space under the BCK/BCI-homomorphisms and the linear transformation of linear spaces, respectively. 2. P RELIMINARIES A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki and was extensively investigated by several researchers. An algebra X := (X; ∗, 0) of type (2, 0) is called a BCI-algebra if it satisfies the following conditions: (I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), (III) (∀x ∈ X) (x ∗ x = 0), (IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y). If a BCI-algebra X satisfies the following identity: (V) (∀x ∈ X) (0 ∗ x = 0), then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following conditions: (a1) (∀x ∈ X) (x ∗ 0 = x), (a2) (∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x), (a3) (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y), (a4) (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) where x ≤ y if and only if x ∗ y = 0. Note that (X , ≤) is a partially ordered set (see [14]). A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. We refer the reader to the books [7, 14] for further information regarding BCK/BCIalgebras. 3. H YBRID STRUCTURES In this paper, we shall use convensions I for the unit interval, L for the set of parameters and P(U ) for the power set of an initial universe set U . Definition 3.1. A hybrid structure in L over U is defined to be a mapping f˜λ := (f˜, λ) : L → P(U ) × I, x 7→ (f˜(x), λ(x)) where f˜ : L → P(U ) and λ : L → I are mappings. Example 3.2. Let U = {h1 , h2 , h3 , h4 , h5 , h6 } be the set of six wooden houses under consideration and let L be the set of parameters which consist of “cheap”, “expensive”, “beautiful” and “in good location”. Consider a hybrid structure in L over U which is given in Table 1. TABLE 1. Tabular representation of the hybrid structure set f˜λ L cheap expensive beautiful in good location Then f˜λ is a hybrid structure in L over U . f˜ {h1 , h3 } {h2 , h5 } {h2 , h3 , h5 } {h2 , h4 } λ 0.3 0.7 0.6 0.5 HYBRID STRUCTURES AND APPLICATIONS 13 Let us denote by H(L) the set of all hybrid structures in L over U . We define an order ≪ in H(L) as follows: ˜ g̃, λ γ (3.1) ∀f˜λ , g̃γ ∈ H(L) f˜λ ≪ g̃γ ⇔ f˜ ⊆ ˜ g̃ means that f˜(x) ⊆ g̃(x) and λ γ means that λ(x) ≥ γ(x) for all x ∈ L. where f˜ ⊆ Note that (H(L), ≪) is a poset. Definition 3.3. Let f˜λ be a hybrid structure in L over U . Then the sets n o f˜λ [α, t] := x ∈ L | f˜(x) ⊇ α, λ(x) ≤ t , n o f˜λ (α, t] := x ∈ L | f˜(x) ) α, λ(x) ≤ t , n o f˜λ [α, t) := x ∈ L | f˜(x) ⊇ α, λ(x) < t , n o f˜λ (α, t) := x ∈ L | f˜(x) ) α, λ(x) < t are called the [α, t]-hybrid cut, (α, t]-hybrid cut, [α, t)-hybrid cut, and (α, t)-hybrid cut of f˜λ , respectively, where α ∈ P(U ) and t ∈ I. Obviously, f˜λ (α, t) ⊆ f˜λ (α, t] ⊆ f˜λ [α, t] and f˜λ (α, t) ⊆ f˜λ [α, t) ⊆ f˜λ [α, t]. Definition 3.4. Let f˜λ and g̃γ be hybrid structures in L over U . Then the hybrid intersection of f˜λ and g̃γ is denoted by f˜λ ⋓ g̃γ and is defined to be a hybrid structure ˜ g̃)(x), (λ ∨ γ)(x) f˜λ ⋓ g̃γ : L → P(U ) × I, x 7→ (f˜ ∩ for all x ∈ L, where ˜ g̃ : L → P(U ), x 7→ f˜(x) ∩ g̃(x), f˜ ∩ _ λ ∨ γ : L → I, x 7→ {λ(x), γ(x)} . (3.2) Definition 3.5. Let f˜λ and g̃γ be hybrid structures in L over U . Then the hybrid union of f˜λ and g̃γ is denoted by f˜λ ⋒ g̃γ and is defined to be a hybrid structure ˜ g̃)(x), (λ ∧ γ)(x) , f˜λ ⋒ g̃γ : L → P(U ) × I, x 7→ (f˜ ∪ where ˜ g̃ : L → P(U ), x 7→ f˜(x) ∪ g̃(x), f˜ ∪ ^ λ ∧ γ : L → I, x 7→ {λ(x), γ(x)} . (3.3) Proposition 3.1. Let f˜λ and g̃γ be two hybrid structures in L over U . For any α, β ∈ P(U ) and s, t ∈ I, we have the following properties: (1) If α ⊆ β and t ≤ s, then f˜λ [β, t] ⊆ f˜λ [α, s]. (2) If α = ∅ and t = 1, then f˜λ [α, t] = L. ˜ ˜ (3) If fλ ≪ g̃γ , then fλ [α, t] ⊆ g̃γ [α, t]. (4) f˜λ ⋓ g̃γ [α, t] = f˜λ [α, t] ∩ g̃γ [α, t]. (5) f˜λ ⋒ g̃γ [α, t] ⊇ f˜λ [α, t] ∪ g̃γ [α, t]. Proof. (1) Assume that α ⊆ β and t ≤ s, and let x ∈ f˜λ [β, t]. Then f˜(x) ⊇ β ⊇ α and λ(x) ≤ t ≤ s, which shows that x ∈ f˜λ [α, s]. Thus f˜λ [β, t] ⊆ f˜λ [α, s]. 14 Y. B. JUN, S. Z. SONG AND G. MUHIUDDIN (2) It is clear. (3) Assume that f˜λ ≪ g̃γ and let x ∈ f˜λ [α, t]. Then g̃(x) ⊇ f˜(x) ⊇ α and γ(x) ≤ λ(x) ≤ t. Hence x ∈ g̃γ [α, t], and so f˜λ [α, t] ⊆ g̃γ [α, t]. (4) For any x ∈ L, we have ˜ g̃ (x) ⊇ α, (λ ∨ γ) (x) ≤ t x ∈ f˜λ ⋓ g̃γ [α, t] ⇔ f˜ ∩ _ ⇔ f˜(x) ∩ g̃(x) ⊇ α, {λ(x), γ(x)} ≤ t ⇔ f˜(x) ⊇ α, g̃(x) ⊇ α, λ(x) ≤ t, γ(x) ≤ t ⇔ x ∈ f˜λ [α, t], x ∈ g̃γ [α, t] ⇔ x ∈ f˜λ [α, t] ∩ g̃γ [α, t]. (5) Since f˜λ ≪ f˜λ ⋒ g̃γ and g̃γ ≪ f˜λ ⋒ g̃γ , it follows from (3) that f˜λ [α, t] ⊆ f˜λ ⋒ g̃γ [α, t] and g̃γ [α, t] ⊆ f˜λ ⋒ g̃γ [α, t]. Therefore f˜λ [α, t] ∪ g̃γ [α, t] ⊆ f˜λ ⋒ g̃γ [α, t]. 4. A PPLICATIONS TO BCK/BCI - ALGEBRAS Definition 4.1. Let L be a BCK/BCI-algebra. A hybrid structure f˜λ in L over U is called a hybrid subalgebra of L over U if the following assertions are valid: f˜(x ∗ y) ⊇ W f˜(x) ∩ f˜(y), (∀x, y ∈ L) . (4.1) λ(x ∗ y) ≤ {λ(x), λ(y)} Example 4.2. Suppose that there are five houses in the initial universe set U given by U = {h1 , h2 , h3 , h4 , h5 } . Let a set of parameters L = {e0 , e1 , e2 , e3 } be a set of status of houses in which e0 stands for the parameter “beautiful”, e1 stands for the parameter “cheap”, e2 stands for the parameter “in good location”, e3 stands for the parameter “in green surroundings, with the Cayley table in Table 2. TABLE 2. Cayley table of the binary operation ∗ ∗ e0 e1 e2 e3 e0 e0 e1 e2 e3 e1 e0 e0 e2 e3 e2 e0 e1 e0 e3 e3 e0 e1 e2 e0 Then (L, ∗, e0 ) is a BCK-algebra. Let f˜λ be a hybrid structure in L over U which is given by Table 3. It is routine to verify that f˜λ is a hybrid subalgebra of L over U . HYBRID STRUCTURES AND APPLICATIONS 15 TABLE 3. Tabular representation of the hybrid structure f˜λ f˜ {h1 , h2 , h3 , h4 , h5 } {h1 , h4 , h5 } {h1 , h3 , h4 } {h2 , h3 , h5 } L e0 e1 e2 e3 λ 0.2 0.5 0.7 0.3 Lemma 4.1. Every hybrid subalgebra f˜λ of a BCK/BCI-algebra L over U satisfies: (∀x ∈ L) f˜(x) ⊆ f˜(0), λ(x) ≥ λ(0) . (4.2) Proof. Note that x ∗ x = 0 for all x ∈ L. Hence f˜(0) = f˜(x ∗ x) ⊇ f˜(x) ∩ f˜(x) = f˜(x), _ λ(0) = λ(x ∗ x) ≤ {λ(x), λ(x)} = λ(x) for all x ∈ L. Proposition 4.2. For a hybrid subalgebra f˜λ of a BCK/BCI-algebra L over U , the following are equivalent: (1) (∀x, y ∈ L) f˜(x ∗ y) ⊇ f˜(y), λ(x ∗ y) ≤ λ(y) . (2) (∀x ∈ L) f˜(0) = f˜(x), λ(0) = λ(x) . Proof. If we take y = 0 in (1), then f˜(0) ⊆ f˜(x ∗ 0) = f˜(x) and λ(0) ≥ λ(x ∗ 0) = λ(x) for all x ∈ L. Combining this and Lemma 4.1, we have f˜(0) = f˜(x) and λ(0) = λ(x) for all x ∈ L. Conversely, assume that (2) is valid. Then f˜(y) = f˜(0) ∩ f˜(y) = f˜(x) ∩ f˜(y) ⊆ f˜(x ∗ y), _ _ λ(y) = {λ(0), λ(y)} = {λ(x), λ(y)} ≥ λ(x ∗ y) for all x, y ∈ L. Proposition 4.3. In a BCI-algebra L, every hybrid subalgebra f˜λ of L over U satisfies the following condition: f˜(x ∗ (0 ∗ y)) ⊇ W f˜(x) ∩ f˜(y), (∀x, y ∈ L) . (4.3) λ(x ∗ (0 ∗ y)) ≤ {λ(x), λ(y)} Proof. Using (4.1) and (4.2), we have f˜(x ∗ (0 ∗ y)) ⊇ f˜(x) ∩ f˜(0 ∗ y) ⊇ f˜(x) ∩ f˜(0) ∩ f˜(y) = f˜(x) ∩ f˜(y) and λ(x ∗ (0 ∗ y)) ≤ for all x, y ∈ L. _ {λ(x), λ(0 ∗ y)} ≤ _ {λ(x), _ {λ(0), λ(y)}} = _ {λ(x), λ(y)} Theorem 4.4. Let L be a BCK/BCI-algebra. For a hybrid structure f˜λ in L over U , the following are equivalent: (1) f˜λ is a hybrid subalgebra of L over U . 16 Y. B. JUN, S. Z. SONG AND G. MUHIUDDIN (2) For any α ∈ P(U ) and t ∈ I, the nonempty sets f˜λ (α) := {x ∈ L | α ⊆ f˜(x)} and f˜λ (t) := {x ∈ L | λ(x) ≤ t} are subalgebras of L. Proof. Assume that f˜λ is a hybrid subalgebra of L over U . Let α ∈ P(U ) and t ∈ I be such that f˜λ (α) 6= ∅ and f˜λ (t) 6= ∅. If x, y ∈ f˜λ (α) ∩ f˜λ (t), then α ⊆ f˜(x), α ⊆ f˜(y), λ(x) ≤ t and λ(y) ≤ t. It follows from (4.1) that f˜(x ∗ y) ⊇ f˜(x) ∩ f˜(y) ⊇ α and λ(x ∗ y) ≤ _ {λ(x), λ(y)} ≤ t. Hence x ∗ y ∈ f˜λ (α) ∩ f˜λ (t), and so f˜λ (α) and f˜λ (t) are subalgebras of L. Conversely, suppose that the second assertion is valid. Let x, y ∈ L be such that f˜(x) = αx and f˜(y) = αy . Taking α = αx ∩ αy implies that x, y ∈ f˜λ (α), and so x ∗ y ∈ f˜λ (α). Hence f˜(x ∗ y) ⊇ α = αx ∩ αy = f˜(x) ∩ f˜(y). W For any x, y ∈ L, let t := {λ(x), λ(y)}. Then x, y ∈ f˜λ (t), and so x ∗ y ∈ f˜λ (t). It follows that _ λ(x ∗ y) ≤ t = {λ(x), λ(y)}. Hence f˜λ is a hybrid subalgebra of L over U . Corollary 4.5. Let L be a BCK/BCI-algebra. If f˜λ is a hybrid subalgebra of L over U , then the nonempty [α, t]-hybrid cut of f˜λ is a subalgebra of L for all α ∈ P(U ) and t ∈ I. Proof. Straightforward. Theorem 4.6. If f˜λ is a hybrid subalgebra of a BCK/BCI-algebra L over U , then the set Ω := {x ∈ L | f˜(x) ∩ α 6= ∅, λ(x) ≤ t} is a subalgebra of L for all (α, t) ∈ P(U ) × I with α 6= ∅ whenever it is nonempty. Proof. Let (α, t) ∈ P(U ) × I be such that Ω 6= ∅ = 6 α. Let x, y ∈ Ω. Then f˜(x) ∩ α 6= ˜ ∅= 6 f (y) ∩ α, λ(x) ≤ t and λ(y) ≤ t. It follows from (4.1) that f˜(x ∗ y) ∩ α ⊇ (f˜(x) ∩ f˜(y)) ∩ α = (f˜(x) ∩ α) ∩ (f˜(y) ∩ α) 6= ∅ and λ(x ∗ y) ≤ W {λ(x), λ(y)} ≤ t. Hence x ∗ y ∈ Ω, and so Ω is a subalgebra of L. Theorem 4.7. Let L be a BCK/BCI-algebra. If f˜λ and g̃γ are hybrid subalgebras of L over U , then so is the hybrid intersection f˜λ ⋓ g̃γ . Proof. For any x, y ∈ L, we have ˜ g̃ (x ∗ y) = f˜(x ∗ y) ∩ g̃(x ∗ y) ⊇ f˜(x) ∩ f˜(y) ∩ (g̃(x) ∩ g̃(y)) f˜ ∩ ˜ g̃ (y) ˜ g̃ (x) ∩ f˜ ∩ = f˜(x) ∩ g̃(x) ∩ f˜(y) ∩ g̃(y) = f˜ ∩ HYBRID STRUCTURES AND APPLICATIONS 17 and (λ ∨ γ) (x ∗ y) = _ {λ(x ∗ y), γ(x ∗ y)} o _ n_ _ ≤ {λ(x), λ(y)} , {γ(x), γ(y)} o _ n_ _ = {λ(x), γ(x)} , {λ(y), γ(y)} _ = {(λ ∨ γ) (x), (λ ∨ γ) (y)} . Hence f˜λ ⋓ g̃γ is a hybrid subalgebra of L over U . The following example shows that the hybrid union of hybrid subalgebras may not be a hybrid subalgebra. Example 4.3. Suppose that there are five patients in a hospital given by U = {p1 , p2 , p3 , p4 , p5 } . As a set of parameters, we consider L = {c, e, g, h, t} which is a set of status of patients where c stands for the parameter “cough”, e stands for the parameter “eye disease”, g stands for the parameter “gastric cancer”, h stands for the parameter “headache”, t stands for the parameter “toothache”. We define a binary operation on L by the Cayley table in Table 4. TABLE 4. Cayley table of the binary operation ∗ ∗ c e g h t c c e g h t e c c g h h g c e c h t h h t h c e t h h h c c Then (L, ∗, c) is a BCI-algebra. Let f˜λ and g̃γ be hybrid structures in L over U which are given by Table 5 and Table 6, respectively. Then the hybrid union f˜λ ⋒ g̃γ of f˜λ and g̃γ is represented by the by Table 7. TABLE 5. Tabular representation of the hybrid structure f˜λ L c e g h t f˜ U {p1 , p2 , p3 , p4 } {p2 , p4 } {p2 , p4 } {p2 , p4 } λ 0.4 0.4 0.7 0.7 0.7 18 Y. B. JUN, S. Z. SONG AND G. MUHIUDDIN TABLE 6. Tabular representation of the hybrid structure g̃γ L c e g h t g̃ U {p2 , p4 } {p1 , p2 , p3 , p4 } {p2 , p3 , p4 } {p2 , p4 } γ 0.2 0.8 0.4 0.6 0.8 TABLE 7. Tabular representation of the hybrid structure f˜λ ⋒ g̃γ L c e g h t ˜ g̃ f˜ ∪ U {p1 , p2 , p3 , p4 } {p1 , p2 , p3 , p4 } {p2 , p3 , p4 } {p2 , p4 } λ∧γ 0.2 0.4 0.4 0.6 0.7 By routine calculations, we know that f˜λ and g̃γ are hybrid subalgebras of L over U . But, the hybrid union f˜λ ⋒ g̃γ is not a hybrid subalgebra of L over U since ˜ g̃ (h) ˜ g̃ (e) ∩ f˜ ∪ ˜ g̃ (t) = {p2 , p4 } + {p2 , p3 , p4 } = f˜ ∪ ˜ g̃ (e ∗ h) = f˜ ∪ f˜ ∪ and/or (λ ∧ γ) (e ∗ h) = (λ ∧ γ) (t) = 0.7 0.6 = _ {(λ ∧ γ) (e), (λ ∧ γ) (h)} . For any hybrid structure f˜λ in L over U , let f˜λ∗ := f˜∗ , λ∗ be a hybrid structure in L over U defined by f˜(x) if x ∈ f˜λ (α), f˜∗ : L → P(U ), x 7→ β otherwise, λ(x) if x ∈ f˜λ (t), λ∗ : L → I, x 7→ s otherwise, where α, β ∈ P(U ) and s, t ∈ I with β ( f˜(x) and s > λ(x). Theorem 4.8. Let L be a BCK/BCI-algebra. If f˜λ is a hybrid subalgebra of L over U , then so is f˜λ∗ . Proof. Assume that f˜λ is a hybrid subalgebra of a BCK/BCI-algebra L over U . Then f˜λ (α) and f˜λ (t) are subalgebras of L for all α ∈ P(U ) and t ∈ I provided that they are nonempty by Theorem 4.4. Let x, y ∈ L. If x, y ∈ f˜λ (α), then x ∗ y ∈ f˜λ (α). Thus f˜∗ (x ∗ y) = f˜(x ∗ y) ⊇ f˜(x) ∩ f˜(y) = f˜∗ (x) ∩ f˜∗ (y). If x ∈ / f˜λ (α) or y ∈ / f˜λ (α), then f˜∗ (x) = β or f˜∗ (y) = β. Hence f˜∗ (x ∗ y) ⊇ β = f˜∗ (x) ∩ f˜∗ (y). Now, if x, y ∈ f˜λ (t), then x ∗ y ∈ f˜λ (t). Thus _ _ λ∗ (x ∗ y) = λ(x ∗ y) ≤ {λ(x), λ(y)} = {λ∗ (x), λ∗ (y)} . HYBRID STRUCTURES AND APPLICATIONS 19 If x ∈ / f˜λ (t) or y ∈ / f˜λ (t), then λ∗ (x) = s or λ∗ (y) = s. Hence _ λ∗ (x ∗ y) ≤ s = {λ∗ (x), λ∗ (y)} . Therefore f˜λ∗ is a hybrid subalgebra of L over U . The following example shows that the converse of Theorem 4.8 is not true in general. Example 4.4. Suppose that there are ten houses in the initial universe set U given by U = {h1 , h2 , h3 , h4 , h5 , h6 , h7 , h8 , h9 , h10 } . Let a set of parameters L = {e0 , e1 , e2 , e3 } be a set of status of houses in which e0 e1 e2 e3 stands for the parameter “beautiful”, stands for the parameter “cheap”, stands for the parameter “in good location”, stands for the parameter “in green surroundings, with the Cayley table in Table 8. TABLE 8. Cayley table of the binary operation ∗ ∗ e0 e1 e2 e3 e0 e0 e1 e2 e3 e1 e1 e0 e3 e2 e2 e2 e3 e0 e1 e3 e3 e2 e1 e0 Then (L, ∗, e0 ) is a BCI-algebra. Let f˜λ be a hybrid structure in L over U which is given by Table 9. TABLE 9. Tabular representation of the hybrid structure f˜λ f˜ U {h2 , h4 , h6 , h8 , h10 } {h3 , h6 , h9 } {h8 } L e0 e1 e2 e3 λ 0.3 0.7 0.9 0.9 ˜ Then f˜λ (α) = {e0, e1 } for α = {h2 , h4 , h6 , h8 , h10 }, and fλ (t) = {e0 , e1 } for t = 0.7. ∗ ∗ ∗ ˜ ˜ Let f := f , λ be a hybrid structure in L over U defined by λ f˜∗ : L → P(U ), x 7→ λ∗ : L → I, x 7→ λ(x) 1 f˜(x) ∅ if x ∈ f˜λ (α), otherwise, if x ∈ f˜λ (t), otherwise, 20 Y. B. JUN, S. Z. SONG AND G. MUHIUDDIN that is,  if x = e0 ,  U {h2 , h4 , h6 , h8 , h10 } if x = e1 , f˜∗ : L → P(U ), x 7→  ∅ if x ∈ {e2 , e3 },   0.3 if x = e0 , 0.7 if x = e1 , λ∗ : L → I, x 7→  1 if x ∈ {e2 , e3 }. It is routine to verify that f˜λ∗ := f˜∗ , λ∗ is a hybrid subalgebra of L over U . But f˜λ is not a hybrid subalgebra of L over U since f˜(e1 ) ∩ f˜(e2 ) = {h6 } * {h8 } = f˜(e3 ) = f˜(e1 ∗ e2 ). Let θ : L → M be a mapping from a set L to a set M . For a hybrid structure g̃γ in M over U , consider a hybrid structure θ−1 (g̃γ ) := θ−1 (g̃), θ−1 (γ) in L over U where θ−1 (g̃)(x) = g̃(θ(x)) and θ−1 (γ)(x) = γ(θ(x)) for all x ∈ L. We say that θ−1 (g̃γ ) is the hybrid preimage of g̃γ under θ. For a hybrid structure f˜ U , the hybrid image of λ in L over ˜ ˜ ˜ fλ under θ is defined to be a hybrid structure θ(fλ ) := θ(f ), θ(λ) in M over U where  S ˜  f (x) if θ−1 (y) 6= ∅, ˜ x∈θ −1 (y) θ(f )(y) =  ∅ otherwise, V ( λ(x) if θ−1 (y) 6= ∅, −1 (y) x∈θ θ(λ)(y) = 1 otherwise, for every y ∈ M . Theorem 4.9. Every homomorphic hybrid preimage of a hybrid subalgebra is also a hybrid subalgebra. Proof. Let θ : L → M be a homomorphism of BCK/BCI-algebras. Let g̃γ is a hybrid subalgebra of M over U and let x, y ∈ L. Then θ−1 (g̃)(x ∗ y) = g̃(θ(x ∗ y)) = g̃(θ(x) ∗ θ(y)) ⊇ g̃(θ(x)) ∩ g̃(θ(y)) = θ−1 (g̃)(x) ∩ θ−1 (g̃)(y) and θ−1 (γ)(x ∗ y) = γ(θ(x ∗ y)) = γ(θ(x) ∗ θ(y)) _ ≤ {γ(θ(x)), γ(θ(y))} _ = {θ−1 (γ)(x), θ−1 (γ)(y)}. Therefore θ−1 (g̃γ ) is a hybrid subalgebra of L over U . For an onto homomorphism θ : L → M of BCK/BCI-algebras, let θ−1 (g̃γ ) := −1 θ (g̃), θ (γ) be a hybrid subalgebra of L over U where g̃γ is a hybrid structure in M over U . Let a, b ∈ M . Then θ(x) = a and θ(y) = b for some x, y ∈ L since θ is onto. −1 HYBRID STRUCTURES AND APPLICATIONS 21 Hence g̃(a ∗ b) = g̃(θ(x) ∗ θ(y)) = g̃(θ(x ∗ y)) = θ−1 (g̃)(x ∗ y) ⊇ θ−1 (g̃)(x) ∩ θ−1 (g̃)(y) = g̃(θ(x)) ∩ g̃(θ(y)) = g̃(a) ∩ g̃(b) and γ(a ∗ b) = γ(θ(x) ∗ θ(y)) = γ(θ(x ∗ y)) = θ−1 (γ)(x ∗ y) _ _ ≤ {θ−1 (γ)(x), θ−1 (γ)(y)} = {γ(θ(x)), γ(θ(y))} _ = {γ(a), γ(b)}. Therefore we have the following theorem. Theorem 4.10. Let θ : L → M be an onto homomorphism of BCK/BCI-algebras. For every hybrid structure g̃γ in M over U , if the preimage θ−1 (g̃γ ) of g̃γ under θ is a hybrid subalgebra of L over U , then g̃γ is a hybrid subalgebra of M over U . 5. A PPLICATIONS TO LINEAR SPACES Definition 5.1. Let L be a field. A hybrid structure f˜λ in L over U is called a hybrid field of L over U if _ (∀a, b ∈ L) f˜(a + b) ⊇ f˜(a) ∩ f˜(b), λ(a + b) ≤ {λ(a), λ(b)} , (5.1) (5.2) (∀a ∈ L) f˜(−a) ⊇ f˜(a), λ(−a) ≤ λ(a) , _ (5.3) (∀a, b ∈ L) f˜(ab) ⊇ f˜(a) ∩ f˜(b), λ(ab) ≤ {λ(a), λ(b)} , (∀a ∈ L) a 6= 0 ⇒ f˜(a−1 ) ⊇ f˜(a), λ(a−1 ) ≤ λ(a) . (5.4) Proposition 5.1. If f˜λ is a hybrid field of a field L over U , then (i) (∀a ∈ L) f˜(a) ⊆ f˜(0), λ(a) ≥ λ(0) , (ii) (∀a ∈ L) a 6= 0 ⇒ f˜(a) ⊆ f˜(1), λ(a) ≥ λ(1) , (iii) f˜(1) ⊆ f˜(0), λ(1) ≥ λ(0). Proof. (i) For every a ∈WL, we have f˜(0) = f˜(a + (−a)) ⊇ f˜(a) ∩ f˜(−a) = f˜(a) and λ(0) = λ(a + (−a)) ≤ {λ(a), λ(−a)} = λ(a). (ii) Let a ∈ L and f˜(1) = f˜(aa−1 ) ⊇ f˜(a) ∩ f˜(a−1 ) = f˜(a) and W a 6= 0. Then −1 −1 λ(1) = λ(aa ) ≤ {λ(a), λ(a )} = λ(a). (iii) It is by (i). Definition 5.2. Let f˜λ be a hybrid field of a field L over U , and let Y be a linear space over L. A hybrid structure g̃γ in Y over U is called a hybrid linear space over (f˜λ , L) if the following conditions hold. _ (∀x, y ∈ Y ) g̃(x + y) ⊇ g̃(x) ∩ g̃(y), γ(x + y) ≤ {γ(x), γ(y)} , (5.5) (∀x ∈ Y ) (g̃(−x) ⊇ g̃(x), γ(−x) ≤ γ(x)) , (5.6) _ (∀a ∈ L)(∀x ∈ Y ) g̃(ax) ⊇ f˜(a) ∩ g̃(x), γ(ax) ≤ {λ(a), γ(x)} , (5.7) f˜(1) ⊇ g̃(0), λ(1) ≤ γ(0). Proposition 5.2. If g̃γ is a hybrid linear space over (f˜λ , L), then (5.8) 22 Y. B. JUN, S. Z. SONG AND G. MUHIUDDIN (i) f˜(0) ⊇ g̃(0), λ(0) ≤ γ(0), (ii) (∀x ∈ Y ) (g̃(0) ⊇ g̃(x), γ(0) ≤ γ(x)), (iii) (∀x ∈ Y ) f˜(0) ⊇ g̃(x), λ(0) ≤ γ(x) . Proof. Straightforward. Theorem 5.3. Let f˜λ be a hybrid field of a field L over U , and let Y be a linear space over L. A hybrid structure g̃γ in Y over U is a hybrid linear space over (f˜λ , L) if and only if W (i) g̃(ax+by) ⊇ f˜(a)∩g̃(x)∩f˜(b)∩g̃(y) and γ(ax+by) ≤ {λ(a), γ(x), λ(b), γ(y)} (ii) f˜(1) ⊇ g̃(x) and λ(1) ≤ γ(x) for all x, y ∈ Y and a, b ∈ L. Proof. Assume that g̃γ is a hybrid linear space over (f˜λ , L). Let a, b ∈ L and x, y ∈ Y . Then g̃(ax + by) ⊇ g̃(ax) ∩ g̃(by) ⊇ (f˜(a) ∩ g̃(x)) ∩ (f˜(b) ∩ g̃(y)) = f˜(a) ∩ g̃(x) ∩ f˜(b) ∩ g̃(y) and γ(ax + by) ≤ = _ _ {γ(ax), γ(by)} ≤ _ n_ {λ(a), γ(x)}, {λ(a), γ(x), λ(b), γ(y)} . _ o {λ(b), γ(y)} The second result is induced by (5.8) and Proposition 5.2(ii). Conversely, suppose that (i) and (ii) are valid. Then g̃(x + y) = g̃(1x + 1y) ⊇ f˜(1) ∩ g̃(x) ∩ f˜(1) ∩ g̃(y) = g̃(x) ∩ g̃(y) and γ(x + y) = γ(1x + 1y) ≤ _ {λ(1), γ(x), λ(1), γ(y)} = for all x, y ∈ Y . Since f˜λ is a hybrid field of L over U , f˜(0) ⊇ f˜(1) ⊇ g̃(x), λ(0) ≤ λ(1) ≤ γ(x), _ {γ(x), γ(y)} f˜(−1) ⊇ f˜(1) ⊇ g̃(x), λ(−1) ≤ λ(1) ≤ γ(x) for all x ∈ Y . Hence g̃(−x) = g̃(0x + (−1)x) ⊇ f˜(0) ∩ g̃(x) ∩ f˜(−1) ∩ g̃(x) = g̃(x) ∩ g̃(x) = g̃(x) and _ γ(−x) = γ(0x + (−1)x) ≤ {λ(0), γ(x), λ(−1), γ(x)} _ = {γ(x), γ(x)} = γ(x). For any a ∈ L and x ∈ Y , we have g̃(ax) = g̃(0x + ax) ⊇ f˜(0) ∩ g̃(x) ∩ f˜(a) ∩ g̃(x) = f˜(a) ∩ g̃(x) and γ(ax) = γ(0x + ax) ≤ _ {λ(0), γ(x), λ(a), γ(x)} = _ {λ(a), γ(x)}. Obviously, f˜(1) ⊇ g̃(0) and λ(1) ≤ γ(0). Therefore g̃γ is a hybrid linear space over (f˜λ , L). HYBRID STRUCTURES AND APPLICATIONS 23 Theorem 5.4. Let θ be a linear transformation of Y into Z where Y and Z are linear spaces over a field L. If f˜λ is a hybrid field of L over U and h̃τ is a hybrid linear space over −1 −1 −1 ˜ (fλ , L) in Z, then θ (h̃τ ) := θ (h̃), θ (τ ) is a hybrid linear space over (f˜λ , L) in Y. Proof. Let x, y ∈ Y and a, b ∈ L. Then θ−1 (h̃)(ax + by) = h̃(θ(ax + by)) = h̃(aθ(x) + bθ(y)) ⊇ f˜(a) ∩ h̃(θ(x)) ∩ f˜(b) ∩ h̃(θ(y)) = f˜(a) ∩ θ−1 (h̃)(x) ∩ f˜(b) ∩ θ−1 (h̃)(y) and θ−1 (τ )(ax + by) = τ (θ(ax + by)) = τ (aθ(x) + bθ(y)) _ ≤ {λ(a), τ (θ(x)), λ(b), τ (θ(y))} _ = λ(a), θ−1 (τ )(x), λ(b), θ−1 (τ )(y) . −1 −1 −1 Obviously, f˜(1) ⊇ θ (h̃)(x) and λ(1) ≤ θ (τ )(x) for all x ∈ Y . Therefore θ (h̃τ ) := θ−1 (h̃), θ−1 (τ ) is a hybrid linear space over (f˜λ , L) in Y by Theorem 5.3. Theorem 5.5. Let θ be a linear transformation of Y into Z where Y and Z are linear spaces over a field L. If f˜λ is a hybrid field of L over U and g̃γ is a hybrid linear space over (f˜λ , L) in Y , then θ(g̃γ ) is a hybrid linear space over (f˜λ , L) in Z. Proof. Let a, b ∈ L and u, v ∈ Z. If θ−1 (u) = ∅ or θ−1 (v) = ∅, then the condition (i) of Theorem 5.3 is satisfied. Assume that θ−1 (u) 6= ∅ 6= θ−1 (v). Then θ−1 (au + bv) 6= ∅. Let r ∈ θ−1 (u) and s ∈ θ−1 (v). Then θ(ar + bs) = aθ(r) + bθ(s) = au + bv, and so [ θ(g̃)(au + bv) = g̃(w) w∈θ −1 (au+bv) [ ⊇ g̃(ar + bs) r∈θ −1 (u), s∈θ −1 (v) ⊇ [ r∈θ −1 (u), s∈θ −1 (v)  = f˜(a) ∩ [ r∈θ −1 (u) f˜(a) ∩ g̃(r) ∩ f˜(b) ∩ g̃(s)   g̃(r) ∩ f˜(b) ∩ s∈θ −1 (v) = f˜(a) ∩ θ(g̃)(u) ∩ f˜(b) ∩ θ(g̃)(v) = f˜(a) ∩ θ(g̃)(u) ∩ f˜(b) ∩ θ(g̃)(v) [  g̃(s) 24 Y. B. JUN, S. Z. SONG AND G. MUHIUDDIN and ^ θ(γ)(au + bv) = γ(w) w∈θ −1 (au+bv) ^ ≤ γ(ar + bs) r∈θ −1 (u), s∈θ −1 (v) ^ ≤ r∈θ −1 (u), s∈θ −1 (v) = =   _ _  _   λ(a), _ ^ {λ(a), γ(r), λ(b), γ(s)} r∈θ −1 (u)    _ λ(b), γ(r) ,   {λ(a), θ(γ)(u), λ(b), θ(γ)(v)} . ^ s∈θ −1 (v) γ(s)    Obviously f˜(1) ⊇ θ(g̃)(x) and λ(1) ≤ θ(γ)(x) for all x ∈ Z. It follows from Theorem 5.3 that θ(g̃γ ) is a hybrid linear space over over (f˜λ , L) in Z. 6. C ONCLUSION In this paper, we introduced the notion of hybrid structure in a set of parameters over an initial universe set, and investigate several properties. Using this notion, we introduce the concepts of a hybrid subalgebra, a hybrid field and a hybrid linear space. We consider the hybrid union and hybrid intersection of hybrid subalgebras in BCK/BCI-algebras. We discuss characterizations of a hybrid subalgebra and a hybrid linear space. Given a hybrid subalgebra, we make a new hybrid subalgebra in BCK/BCI-algebras. We consider the hybrid image and preimage of a hybrid subalgebra and a hybrid linear space under the BCK/BCI-homomorphisms and the linear transformation of linear spaces, respectively. Work is ongoing. Some important issues for future work are: (1) to develop strategies for obtaining more valuable results, (2) to apply these notions and results for studying related notions in other (hyper) algebraic structures, (3) to extend these results, one can further study the hybrid algebraic structures on different algebras such as MTL-algerbas, BL-algebras, MV-algebras, EQ-algebras, R0-algebras and Q-algebras etc. 7. ACKNOWLEDGEMENTS The authors would like to express their sincere thanks to the anonymous referee(s) for a careful checking of the details and for helpful comments. R EFERENCES [1] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl. 59 (2010) 3458–3463. [2] H. Aktaş and N. Çağman, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726–2735. [3] A. O. Atagün and A. Sezgin, Soft substructures of rings, fields and modules, Comput. Math. Appl. 61 (2011) 592–601. [4] N. Çaǧman, F. Çitak and S. Enginoğlu, Soft set theory and uni-int decision making, Eur. J. Oper. Res. 207 (2010) 848–855. [5] N. Çaǧman and S. Enginoğlu, Soft matrix theory and its decision making, Comput. Math. Appl. 59 (2010) 3308–3314. [6] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621–2628. [7] Y. Huang, BCI-algebra, Science Press, Beijing 2006. [8] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413. [9] Y. B. Jun, K. J. Lee and A. Khan, Soft ordered semigroups, Math. Logic Q. 56 (2010) 42–50. HYBRID STRUCTURES AND APPLICATIONS 25 [10] Y. B. Jun, K. J. Lee and C. H. Park, Soft set theory applied to ideals in d-algebras, Comput. Math. Appl. 57 (2009) 367–378. [11] Y. B. Jun, K. J. Lee and J. Zhan, Soft p-ideals of soft BCI-algebras, Comput. Math. Appl. 58 (2009) 2060– 2068. [12] Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform. Sci. 178 (2008) 2466–2475. [13] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077–1083. [14] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa Co. Seoul 1994. [15] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. [16] D. A. Molodtsov, The description of a dependence with the help of soft sets, Journal of Computer and Systems Sciences International 40(6) (2001) 977–984. [17] D. A. Molodtsov, The Theory of Soft Sets, URSS Publishers, Moscow, 2004, (in Russian). [18] D. A. Molodtsov, V. Yu. Leonov, D. V. Kovkov, Soft sets technique and its application, Nechetkie Sistemy i Myagkie Vychisleniya 1(1) (2006) 8–39. [19] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010), 529–539. [20] V. Torra and Y. Narukawa, On hesitant fuzzy sets and decision, in: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378. 1382. [21] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [22] J. Zhan and Y. B. Jun, Soft BL-algebras based on fuzzy sets, Comput. Math. Appl. 59 (2010) 2037–2046. YOUNG BAE J UN D EPARTMENT OF M ATHEMATICS E DUCATION , G YEONGSANG NATIONAL U NIVERSITY, J INJU 660-701, KOREA Email address: skywine@gmail.com S EOK Z UN S ONG D EPARTMENT OF M ATHEMATICS ,, J EJU NATIONAL U NIVERSITY, J EJU 690-756, KOREA Email address: szsong@jejunu.ac.kr G. M UHIUDDIN D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF TABUK , TABUK 71491, S AUDI A RABIA Email address: chishtygm@gmail.com ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 1, Number 1 (2018), 26-37 ISSN: 2582-0818 c http://www.technoskypub.com p-SEMISIMPLE NEUTROSOPHIC QUADRUPLE BCI-ALGEBRAS AND NEUTROSOPHIC QUADRUPLE p-IDEALS G. MUHIUDDIN∗ AND YOUNG BAE JUN A BSTRACT. Characterizations of neutrosophic quadruple BCI-algebra are considered. Conditions for the neutrosophic quadruple BCI-set to be a p-semisimple BCI-algebra are provided. A condition for a subalgebra to be an ideal in neutrosophic quadruple BCIalgebra is given. Conditions for the set N Q(A, B) to be a neutrosophic quadruple closed ideal and neutrosophic quadruple p-ideal are discussed. Characterizations of neutrosophic quadruple p-ideal are considered. 1. I NTRODUCTION As a more general platform that extends the notions of classic set, (intuitionistic) fuzzy set and interval valued (intuitionistic) fuzzy set, the notion of neutrosophic set is developed by Smarandache ([16], [17] and [18]). Neutrosophic algebraic structures in BCK/BCIalgebras are discussed in the papers [3], [7], [8], [9], [10], [12], [14], [15] and [20]. Smarandache [19] considered an entry (i.e., a number, an idea, an object etc.) which is represented by a known part (a) and an unknown part (bT, cI, dF ) where T, I, F have their usual neutrosophic logic meanings and a, b, c, d are real or complex numbers, and then he introduced the concept of neutrosophic quadruple numbers. Neutrosophic quadruple algebraic structures and hyperstructures are discussed in [1] and [2]. Jun et al. [11] used neutrosophic quadruple numbers based on a set, and constructed neutrosophic quadruple BCK/BCI-algebras. They investigated several properties, and considered ideal and positive implicative ideal in neutrosophic quadruple BCK-algebra, and closed ideal in neutrosophic quadruple BCI-algebra. Given subsets A and B of a neutrosophic quadruple BCK/BCI-algebra, they considered sets N Q(A, B) which consists of neutrosophic quadruple BCK/BCI-numbers with a condition. They provided conditions for the set N Q(A, B) to be a (positive implicative) ideal of a neutrosophic quadruple BCK-algebra, and the set N Q(A, B) to be a (closed) ideal of a neutrosophic quadruple BCI-algebra. They gave an example to show that the set {0̃} is not a positive implicative ideal in a neutrosophic quadruple BCK-algebra, and then they considered conditions for the set {0̃} to be a positive implicative ideal in a neutrosophic quadruple BCK-algebra. 2010 Mathematics Subject Classification. 06F35, 03G25, 08A72. Key words and phrases. Neutrosophic quadruple BCK/BCI-number; neutrosophic quadruple BCK/BCIalgebra; neutrosophic quadruple (closed) ideal; neutrosophic quadruple p-ideal. *Corresponding author. 26 p-SEMISIMPLE NEUTROSOPHIC QUADRUPLE BCI-ALGEBRAS 27 In this paper, we consider characterizations of neutrosophic quadruple BCI-algebra, and give conditions for the neutrosophic quadruple BCI-set to be a p-semisimple BCIalgebra. We provide a condition for a subalgebra to be an ideal in neutrosophic quadruple BCI-algebra, and provide conditions for the set N Q(A, B) to be a neutrosophic quadruple closed ideal and neutrosophic quadruple p-ideal. We disuss characterizations of neutrosophic quadruple p-ideal. 2. P RELIMINARIES A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki (see [5] and [6]) and was extensively investigated by several researchers. By a BCI-algebra, we mean a set S with a special element 0 and a binary operation ∗ that satisfies the following conditions: (I) (II) (III) (IV) (∀x, y, z ∈ S) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (∀x, y ∈ S) ((x ∗ (x ∗ y)) ∗ y = 0), (∀x ∈ S) (x ∗ x = 0), (∀x, y ∈ S) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y). If a BCI-algebra S satisfies the following identity: (V) (∀x ∈ S) (0 ∗ x = 0), then S is called a BCK-algebra. Any BCK/BCI-algebra S satisfies the following conditions: (∀x ∈ S) (x ∗ 0 = x) , (∀x, y, z ∈ S) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x) , (2.1) (2.2) (∀x, y, z ∈ S) ((x ∗ y) ∗ z = (x ∗ z) ∗ y) , (2.3) (∀x, y, z ∈ S) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (2.4) where x ≤ y if and only if x ∗ y = 0. Any BCI-algebra S satisfies the following conditions (see [4]): (∀x, y ∈ S)(x ∗ (x ∗ (x ∗ y)) = x ∗ y), (2.5) (∀x, y ∈ S)(0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y)), (2.6) (∀x, y ∈ S)(0 ∗ (0 ∗ (x ∗ y)) = (0 ∗ y) ∗ (0 ∗ x)). (2.7) A BCI-algebra S is said to be p-semisimple (see [4]) if 0 ∗ (0 ∗ x) = x for all x ∈ S. Every p-semisimple BCI-algebra S satisfies (see [4]): (∀x, y, z ∈ S)((x ∗ z) ∗ (y ∗ z) = x ∗ y). (2.8) A BCI-algebra S is p-semisimple if and only if the following assertion is valid. (∀x, y ∈ S)(x ∗ (x ∗ y) = y). (2.9) An element a in a BCI-algebra S is said to be minimal (see [4]) if the following assertion is valid. (∀x ∈ S)(x ∗ a = 0 ⇒ x = a). (2.10) A nonempty subset S of a BCK/BCI-algebra S is called a subalgebra of S if x∗y ∈ S for all x, y ∈ S. A subset I of a BCK/BCI-algebra S is called an ideal of S if it satisfies: 0 ∈ I, (2.11) (∀x ∈ S) (∀y ∈ I) (x ∗ y ∈ I ⇒ x ∈ I) . (2.12) 28 G. MUHIUDDIN AND Y. B. JUN A subset I of a BCI-algebra S is called a closed ideal (see [4]) of S if it is an ideal of S which satisfies: (∀x ∈ S)(x ∈ I ⇒ 0 ∗ x ∈ I). (2.13) A subset I of a BCI-algebra S is called a p-ideal (see [21]) of S if it satisfies (2.11) and (∀x, y, z ∈ S)(y ∈ I, (x ∗ z) ∗ (y ∗ z) ∈ I ⇒ x ∈ I). (2.14) We refer the reader to the books [4, 13] for further information regarding BCK/BCIalgebras, and to the site “http://fs.gallup.unm.edu/neutrosophy.htm” for further information regarding neutrosophic set theory. We consider neutrosophic quadruple numbers based on a set instead of real or complex numbers. Definition 2.1 ([11]). Let S be a set. A neutrosophic quadruple S-number is an ordered quadruple (a, xT, yI, zF ) where a, x, y, z ∈ S and T, I, F have their usual neutrosophic logic meanings. The set of all neutrosophic quadruple S-numbers is denoted by N Q(S), that is, N Q(S) := {(a, xT, yI, zF ) | a, x, y, z ∈ S}, and it is called the neutrosophic quadruple set based on S. If S is a BCK/BCI-algebra, a neutrosophic quadruple S-number is called a neutrosophic quadruple BCK/BCI-number and we say that N Q(S) is the neutrosophic quadruple BCK/BCI-set. Let S be a BCK/BCI-algebra. We define a binary operation ⊙ on N Q(S) by (a, xT, yI, zF ) ⊙ (b, uT, vI, wF ) = (a ∗ b, (x ∗ u)T, (y ∗ v)I, (z ∗ w)F ) for all (a, xT, yI, zF ), (b, uT, vI, wF ) ∈ N Q(S). Given a1 , a2 , a3 , a4 ∈ S, the neutrosophic quadruple BCK/BCI-number (a1 , a2 T, a3 I, a4 F ) is denoted by ã, that is, ã = (a1 , a2 T, a3 I, a4 F ), and the zero neutrosophic quadruple BCK/BCI-number (0, 0T, 0I, 0F ) is denoted by 0̃, that is, 0̃ = (0, 0T, 0I, 0F ). We define an order relation “≪” and the equality “=” on N Q(S) as follows: x̃ ≪ ỹ ⇔ xi ≤ yi for i = 1, 2, 3, 4, x̃ = ỹ ⇔ xi = yi for i = 1, 2, 3, 4 for all x̃, ỹ ∈ N Q(S). It is easy to verify that “≪” is an equivalence relation on N Q(S). Theorem 2.1 ([11]). If S is a BCK/BCI-algebra, then (N Q(S); ⊙, 0̃) is a BCK/BCIalgebra. We say that (N Q(S); ⊙, 0̃) is a neutrosophic quadruple BCK/BCI-algebra, and it is simply denoted by N Q(S). Let S be a BCK/BCI-algebra. Given a, b ∈ S and nonempty subsets A and B of S, consider the sets N Q(a, B) := {(a, aT, yI, zF ) ∈ N Q(S) | y, z ∈ B}, N Q(A, b) := {(a, xT, bI, bF ) ∈ N Q(S) | a, x ∈ A}, N Q(A, B) := {(a, xT, yI, zF ) ∈ N Q(S) | a, x ∈ A; y, z ∈ B}, p-SEMISIMPLE NEUTROSOPHIC QUADRUPLE BCI-ALGEBRAS N Q(A∗ , B) := [ 29 N Q(a, B), a∈A N Q(A, B ∗ ) := [ N Q(A, b), b∈B and N Q(A ∪ B) := N Q(A, 0) ∪ N Q(0, B). The set N Q(A, A) is denoted by N Q(A). 3. p- SEMISIMPLE NEUTROSOPHIC QUADRUPLE BCI - ALGEBRAS AND IDEALS Definition 3.1. Given nonempty subsets A and B of S, if N Q(A, B) is a (closed) ideal (resp., p-ideal) of a neutrosophic quadruple BCI-algebra N Q(S), we say N Q(A, B) is a neutrosophic quadruple (closed) ideal (resp., neutrosophic quadruple p-ideal) of N Q(S). Theorem 3.1. Let N Q(S) be the neutrosophic quadruple set based on a set S. Then (N Q(S); ⊙, 0̃) is a neutrosophic quadruple BCI-algebra if and only if the following assertions are valid. (∀x̃, ỹ, z̃ ∈ N Q(S)) (((x̃ ⊙ ỹ) ⊙ (x̃ ⊙ z̃)) ⊙ (z̃ ⊙ ỹ) = 0̃), (3.1) (∀x̃, ỹ ∈ N Q(S)) (x̃ ⊙ ỹ = 0̃, ỹ ⊙ x̃ = 0̃ ⇒ x̃ = ỹ), (3.2) (∀x̃ ∈ N Q(S)) (x̃ ⊙ 0̃ = x̃). (3.3) Proof. Assume that (N Q(S); ⊙, 0̃) is a neutrosophic quadruple BCI-algebra. Then two conditions (3.1) and (3.2) are clearly true. Note that x̃ ⊙ x̃ = 0̃, (3.4) (x̃ ⊙ (x̃ ⊙ ỹ)) ⊙ ỹ = 0̃ (3.5) (x̃ ⊙ (x̃ ⊙ 0̃)) ⊙ 0̃ = 0̃ (3.6) for all x̃, ỹ ∈ N Q(S). Hence for all x̃ ∈ N Q(S), and it follows from (3.1), (3.4) and (3.6) that 0̃ = ((x̃ ⊙ (x̃ ⊙ 0̃)) ⊙ (x̃ ⊙ x̃)) ⊙ (x̃ ⊙ (x̃ ⊙ 0̃)) = ((x̃ ⊙ (x̃ ⊙ 0̃)) ⊙ 0̃) ⊙ (x̃ ⊙ (x̃ ⊙ 0̃)) = 0̃ ⊙ (x̃ ⊙ (x̃ ⊙ 0̃)). Using (3.2), we have x̃ ⊙ (x̃ ⊙ 0̃) = 0̃ for all x̃ ∈ N Q(S). Also we have (x̃ ⊙ 0̃) ⊙ x̃ = (x̃ ⊙ (x̃ ⊙ x̃)) ⊙ x̃ = 0̃ by (3.4) and (3.5). Therefore (3.3) is valid by using (3.2). Conversely, suppose that the neutrosophic quadruple set N Q(S) based on a set S satisfies three conditions (3.1), (3.2) and (3.3). It is sufficient to show that two conditions (3.4) and (3.5) are true. Let x̃, ỹ ∈ N Q(S). Using (3.3) and (3.1), we have x̃ ⊙ x̃ = (x̃ ⊙ x̃) ⊙ 0̃ = ((x̃ ⊙ 0̃) ⊙ (x̃ ⊙ 0̃)) ⊙ (0̃ ⊙ 0̃) = 0̃ and (x̃ ⊙ (x̃ ⊙ ỹ)) ⊙ ỹ = ((x̃ ⊙ 0̃) ⊙ (x̃ ⊙ ỹ)) ⊙ (ỹ ⊙ 0̃) = 0̃. Therefore (N Q(S); ⊙, 0̃) is a neutrosophic quadruple BCI-algebra. We consider conditions for the neutrosophic quadruple BCI-set N Q(S) to be a psemisimple neutrosophic quadruple BCI-algebra. 30 G. MUHIUDDIN AND Y. B. JUN Theorem 3.2. If S is a p-semisimple BCI-algebra, then (N Q(S); ⊙, 0̃) is a p-semisimple neutrosophic quadruple BCI-algebra. Proof. Let S be a p-semisimple BCI-algebra. Then (N Q(S); ⊙, 0̃) is a neutrosophic quadruple BCI-algebra (see Theorem 2.1). For any x̃ = (x1 , x2 T, x3 I, x4 F ) ∈ N Q(S), we have 0̃ ⊙ (0̃ ⊙ x̃) = (0 ∗ (0 ∗ x1 ), (0 ∗ (0 ∗ x2 ))T, (0 ∗ (0 ∗ x3 ))I, (0 ∗ (0 ∗ x4 ))F ) = (x1 , x2 T, x3 I, x4 F ) = x̃. Hence (N Q(S); ⊙, 0̃) is a p-semisimple neutrosophic quadruple BCI-algebra. Theorem 3.3. If the neutrosophic quadruple set N Q(S) based on a BCI-algebra S satisfies the following assertion (∀x̃ ∈ N Q(S))(0̃ ⊙ x̃ = 0̃ ⇒ x̃ = 0̃), (3.7) then (N Q(S); ⊙, 0̃) is a p-semisimple neutrosophic quadruple BCI-algebra. Proof. By Theorem 2.1, (N Q(S); ⊙, 0̃) is a neutrosophic quadruple BCI-algebra. Thus 0̃ ⊙ (x̃ ⊙ ỹ) = (0̃ ⊙ x̃) ⊙ (0̃ ⊙ ỹ) (3.8) 0̃ ⊙ (0̃ ⊙ (0̃ ⊙ x̃)) = 0̃ ⊙ x̃ (3.9) for all x̃, ỹ ∈ N Q(S). It follows from (3.4) that 0̃ ⊙ (x̃ ⊙ (0̃ ⊙ (0̃ ⊙ x̃))) = (0̃ ⊙ x̃) ⊙ (0̃ ⊙ (0̃ ⊙ (0̃ ⊙ x̃))) = (0̃ ⊙ x̃) ⊙ (0̃ ⊙ x̃) = 0̃. Hence x̃ ⊙ (0̃ ⊙ (0̃ ⊙ x̃)) = 0̃ for all x̃ ∈ N Q(S) by (3.7). Since (0̃ ⊙ (0̃ ⊙ x̃)) ⊙ x̃ = 0̃ for all x̃ ∈ N Q(S), it follows from (3.2) that 0̃ ⊙ (0̃ ⊙ x̃) = x̃ for all x̃ ∈ N Q(S). Therefore (N Q(S); ⊙, 0̃) is a p-semisimple neutrosophic quadruple BCI-algebra. Corollary 3.4. If the neutrosophic quadruple set N Q(S) based on a BCI-algebra S satisfies the following assertion (∀x̃, ỹ ∈ N Q(S))(x̃ ⊙ (0̃ ⊙ ỹ) = ỹ ⊙ (0̃ ⊙ x̃)), (3.10) then (N Q(S); ⊙, 0̃) is a p-semisimple neutrosophic quadruple BCI-algebra. Proof. By Theorem 2.1, (N Q(S); ⊙, 0̃) is a neutrosophic quadruple BCI-algebra. Let x̃ ∈ N Q(S) be such that 0̃ ⊙ x̃ = 0̃. Then x̃ = x̃ ⊙ 0̃ = x̃ ⊙ (0̃ ⊙ 0̃) = 0̃ ⊙ (0̃ ⊙ x̃) = 0̃ ⊙ 0̃ = 0̃ by (3.3), (3.4) and (3.10). It follows from Theorem 3.3 that (N Q(S); ⊙, 0̃) is a p-semisimple neutrosophic quadruple BCI-algebra. In a neutrosophic quadruple BCI-algebra, any subalgebra may not be an ideal as seen in the following example. Example 3.2. Consider a BCI-algebra S = {0, 1, a} with the binary operation ∗, which is given in Table 1. Then the neutrosophic quadruple BCI-algebra N Q(S) has 81 elements. If we take ˜ 11, ˜ 12, ˜ 13, ˜ 14, ˜ 15} ˜ B := {0̃, 1̃, 2̃, 3̃, 4̃, 5̃, 6̃, 7̃, 8̃, 9̃, 10, where 0̃ = (0, 0T, 0I, 0F ), 1̃ = (0, 0T, 0I, aF ), 2̃ = (0, 0T, aI, 0F ), p-SEMISIMPLE NEUTROSOPHIC QUADRUPLE BCI-ALGEBRAS 31 TABLE 1. Cayley table for the binary operation “∗” ∗ 0 1 a 0 0 1 a 1 0 0 a a a a 0 3̃ = (0, 0T, aI, aF ), 4̃ = (0, aT, 0I, 0F ), 5̃ = (0, aT, 0I, aF ), 6̃ = (0, aT, aI, 0F ), 7̃ = (0, aT, aI, aF ), 8̃ = (a, 0T, 0I, 0F ), ˜ = (a, 0T, aI, 0F ), 11 ˜ = (a, 0T, aI, aF ), 9̃ = (a, 0T, 0I, aF ), 10 ˜ ˜ 12 = (a, aT, 0I, 0F ), 13 = (a, aT, 0I, aF ), ˜ = (a, aT, aI, 0F ), 15 ˜ = (a, aT, aI, aF ). 14 Then B is a subalgebra of N Q(S). But it is not an ideal of N Q(S). In fact, if we take x̃ = (1, 1T, 0I, aF ) ∈ N Q(S) then ˜ = (1, 1T, 0I, aF ) ⊙ (a, aT, aI, aF ) = (a, aT, aI, 0F ) = 14 ˜ ∈B x̃ ⊙ 15 But x̃ = (1, 1T, 0I, aF ) ∈ / B. We provide a condition for a subalgebra to be an ideal in neutrosophic quadruple BCIalgebra. Theorem 3.5. If N Q(S) is a neutrosophic quadruple BCI-algebra based on a p-semisimple BCI-algebra S, then every subalgebra of N Q(S) is an ideal of N Q(S). Proof. If S is a p-semisimple BCI-algebra, then (N Q(S); ⊙, 0̃) is a p-semisimple neutrosophic quadruple BCI-algebra by Theorem 3.2. Let N Q(S) be a subalgebra of N Q(S). It is clear that 0̃ ∈ N Q(S). Let x̃, ỹ ∈ N Q(S) be such that x̃⊙ ỹ ∈ N Q(S) and ỹ ∈ N Q(S). Then 0̃ ⊙ ỹ ∈ N Q(S) and (x̃ ⊙ ỹ) ⊙ (0̃ ⊙ ỹ) ∈ N Q(S). Note that ((x̃ ⊙ ỹ) ⊙ (0̃ ⊙ ỹ)) ⊙ x̃ = ((x̃ ⊙ ỹ) ⊙ x̃) ⊙ (0̃ ⊙ ỹ) = ((x̃ ⊙ ỹ) ⊙ (x̃ ⊙ 0̃)) ⊙ (0̃ ⊙ ỹ) = 0̃. Since (N Q(S); ⊙, 0̃) is p-semisimple, we have x̃ = (x̃ ⊙ ỹ) ⊙ (0̃ ⊙ ỹ) ∈ N Q(S) by (3.2). Therefore N Q(S) is an ideal of N Q(S). Lemma 3.6 ([11]). If A and B are (closed) ideals of a BCI-algebra S, then the set N Q(A, B) is a neutrosophic quadruple (closed) ideal of N Q(S). Recall that there exist ideals A and B in a BCI-algebra S such that N Q(A, B) is not a neutrosophic quadruple closed ideal of N Q(S) (see [11, Example 3]). We provide conditions for the set N Q(A, B) to be a neutrosophic quadruple closed ideal of N Q(S). Theorem 3.7. Let A and B be ideals of a BCI-algebra S. Then the set N Q(A, B) is a neutrosophic quadruple closed ideal of N Q(S) if and only if the following assertion is valid. (∀a ∈ A, ∀b ∈ B)(0 ∗ a ∈ A, 0 ∗ b ∈ B). (3.11) Proof. Assume that N Q(A, B) is a neutrosophic quadruple closed ideal of N Q(S) for any ideals A and B of a BCI-algebra S. Let a1 , a2 ∈ A and b1 , b2 ∈ B be such that 32 G. MUHIUDDIN AND Y. B. JUN (a1 , a2 T, b1 I, b2 F ) ∈ N Q(A, B). Then (0 ∗ a1 , (0 ∗ a2 )T, (0 ∗ b1 )I, (0 ∗ b2 )F ) = (0, 0T, 0I, 0F ) ⊙ (a1 , a2 T, b1 I, b2 F ) ∈ N Q(A, B), and so 0 ∗ a1 , 0 ∗ a2 ∈ A and 0 ∗ b1 , 0 ∗ b2 ∈ B. Therefore (3.11) is valid. Conversely, let A and B be ideals of a BCI-algebra S satisfying the condition (3.11). Then A and B are closed ideals of S. It follows from Lemma 3.6 that N Q(A, B) is a neutrosophic quadruple closed ideal of N Q(S). Corollary 3.8. Given an ideal A of a BCI-algebra S, the set N Q(A) is a neutrosophic quadruple closed ideal of N Q(S) if and only if 0 ∗ a ∈ A for all a ∈ A. Theorem 3.9. For any ideals A and B of a BCI-algebra S, let m(A) and m(B) be the set of all minimal elements of A and B with |m(A)| < ∞ and |m(B)| < ∞, respectively. Then the set N Q(A, B) is a neutrosophic quadruple closed ideal of N Q(S). Proof. For any a ∈ A, b ∈ B and n, k ∈ N, let an = 0∗(0∗a)n and bk = 0∗(0∗b)k . Then an ∈ m(A) and bk ∈ m(B). Using (2.6) repeatedly, we have an = 0∗(0∗a)n = 0∗(0∗an ) and bk = 0 ∗ (0 ∗ b)k = 0 ∗ (0 ∗ bk ). Hence an ∗ an = (0 ∗ (0 ∗ an )) ∗ an = (0 ∗ an ) ∗ (0 ∗ an ) = 0 ∈ A and bk ∗ bk = (0 ∗ (0 ∗ bk )) ∗ bk = (0 ∗ bk ) ∗ (0 ∗ bk ) = 0 ∈ B. Since A and B are ideals, it follows that an ∈ A and bk ∈ B. Since |m(A)| < ∞ and |m(B)| < ∞, there exist p, q ∈ N such that an+p = an and bk+q = bk , that is, an ∗ (0 ∗ a)p = an and bk ∗ (0 ∗ b)q = bk . It follows that ap = 0 ∗ (0 ∗ a)p = (an ∗ (0 ∗ a)p ) ∗ an = an ∗ an = 0 and bq = 0 ∗ (0 ∗ b)q = (bk ∗ (0 ∗ b)q ) ∗ bk = bk ∗ bk = 0. Thus ap−1 ∗(0∗a) = 0 and bq−1 ∗(0∗b) = 0, and so 0∗a = ap−1 ∈ A and 0∗b = bq−1 ∈ B. Hence A and B are closed ideals of S. Therefore N Q(A, B) is a neutrosophic quadruple closed ideal of N Q(S). by Lemma 3.6. 4. N EUTROSOPHIC QUADRUPLE p- IDEALS In what follows, let S be a BCI-algebra unless otherwise. Question 1. If A and B are ideals of S, then is N Q(A, B) a neutrosophic quadruple p-ideal of N Q(S)? The following example shows that the answer to Question 1 is negative. Example 4.1. Consider a BCI-algebra S = {0, 1, a, b} with the binary operation ∗, which is given in Table 2. Then the neutrosophic quadruple BCI-algebra N Q(S) has 256 elements. Consider ideals A = {0, 1} and B = {0, a} of S. Note that B = {0, a} is not a p-ideal of S. Then ˜ 11, ˜ 12, ˜ 13, ˜ 14, ˜ 15} ˜ N Q(A, B) = {0̃, 1̃, 2̃, 3̃, 4̃, 5̃, 6̃, 7̃, 8̃, 9̃, 10, is a neutrosophic quadruple ideal of N Q(S) where 0̃ = (0, 0T, 0I, 0F ), 1̃ = (0, 0T, 0I, aF ), 2̃ = (0, 0T, aI, 0F ), 3̃ = (0, 0T, aI, aF ), 4̃ = (0, 1T, 0I, 0F ), 5̃ = (0, 1T, 0I, aF ), p-SEMISIMPLE NEUTROSOPHIC QUADRUPLE BCI-ALGEBRAS 33 TABLE 2. Cayley table for the binary operation “∗” ∗ 0 1 a b 0 0 1 a b 1 0 0 a a a a b 0 1 b a a 0 0 6̃ = (0, 1T, aI, 0F ), 7̃ = (0, 1T, aI, aF ), 8̃ = (1, 0T, 0I, 0F ), ˜ = (1, 0T, aI, 0F ), 11 ˜ = (1, 0T, aI, aF ), 9̃ = (1, 0T, 0I, aF ), 10 ˜ = (1, 1T, 0I, 0F ), 13 ˜ = (1, 1T, 0I, aF ), 12 ˜ = (1, 1T, aI, 0F ), 15 ˜ = (1, 1T, aI, aF ). 14 If we take x̃ = (1, 1T, bI, bF ) ∈ N Q(S) and z̃ = (b, bT, bI, bF ) ∈ N Q(S), then (x̃ ⊙ z̃) ⊙ (7̃ ⊙ z̃) = (a, aT, 0I, 0F ) ⊙ (a, aT, 0I, 0F ) = (0, 0T, 0I, 0F ) = 0̃ ∈ N Q(A, B). But x̃ ∈ / N Q(A, B), and so N Q(A, B) is not a neutrosophic quadruple p-ideal of N Q(S). We provide a condition for the set N Q(A, B) to be a neutrosophic quadruple p-ideal. Theorem 4.1. Let A and B be ideals of S. If S is p-semisimple, then N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S). Proof. If A and B are ideals of S, then N Q(A, B) is an ideal of N Q(S) (see Lemma 3.6), and so 0̃ ∈ N Q(A, B). Let x̃, ỹ, z̃ ∈ N Q(S) be such that (x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃) ∈ N Q(A, B) and ỹ ∈ N Q(A, B). Since S is p-semisimple, it follows from (2.8) that (x1 ∗ y1 , (x2 ∗ y2 )T, (x3 ∗ y3 )I, (x4 ∗ y4 )F ) = ((x1 ∗ z1 ) ∗ (y1 ∗ z1 ), ((x2 ∗ z2 ) ∗ (y2 ∗ z2 ))T, ((x3 ∗ z3 ) ∗ (y3 ∗ z3 ))I, ((x4 ∗ z4 ) ∗ (y4 ∗ z4 ))F ) = (x1 ∗ z1 , (x2 ∗ z2 )T, (x3 ∗ z3 )I, (x4 ∗ z4 )F )⊙ (y1 ∗ z1 , (y2 ∗ z2 )T, (y3 ∗ z3 )I, (y4 ∗ z4 )F ) = ((x1 , x2 T, x3 I, x4 F ) ⊙ (z1 , z2 T, z3 I, z4 F ))⊙ ((y1 , y2 T, y3 I, y4 F ) ⊙ (z1 , z2 T, z3 I, z4 F )) = (x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃) ∈ N Q(A, B). Hence xi ∗ yi ∈ A and xj ∗ yj ∈ B for i = 1, 2 and j = 3, 4. Since y1 , y2 ∈ A and y3 , y4 ∈ B, we have xi ∈ A and xj ∈ B for i = 1, 2 and j = 3, 4. Thus x̃ = (x1 , x2 T, x3 I, x4 F ) ∈ N Q(A, B). Therefore N Q(A, B) is a neutrosophic quadruple pideal of N Q(S). Corollary 4.2. If A is an ideal of a p-semisimple BCI-algebra S, then N Q(A) is a neutrosophic quadruple p-ideal of N Q(S). Corollary 4.3. If a BCI-algebra S satisfies: (∀x, y, z ∈ S)((x ∗ y) ∗ z = x ∗ (y ∗ z)), (4.1) then N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S) for all ideals A and B of S. 34 G. MUHIUDDIN AND Y. B. JUN Proof. Using (2.3) and (4.1), we have y ∗ (x ∗ (x ∗ y)) = (y ∗ x) ∗ (x ∗ y) = (y ∗ (x ∗ y)) ∗ x = ((y ∗ x) ∗ y) ∗ x = (y ∗ x) ∗ (y ∗ x) = 0 for all x, y ∈ S. It follows from (II) and (IV) that x ∗ (x ∗ y) = y. Hence S is psemisimple, and therefore N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S) by Theorem 4.1. Theorem 4.4. If A and B are p-ideals of S, then the set N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S). Proof. Assume that A and B are p-ideals of S. Obviously, 0̃ ∈ N Q(A, B). Let x̃ = (x1 , x2 T, x3 I, x4 F ), ỹ = (y1 , y2 T, y3 I, y4 F ) and z̃ = (z1 , z2 T, z3 I, z4 F ) be elements of N Q(S) such that (x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃) ∈ N Q(A, B) and ỹ ∈ N Q(A, B). Then (x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃) = ((x1 ∗ z1 ) ∗ (y1 ∗ z1 ), ((x2 ∗ z2 ) ∗ (y2 ∗ z2 ))T, ((x3 ∗ z3 ) ∗ (y3 ∗ z3 ))I, ((x4 ∗ z4 ) ∗ (y4 ∗ z4 ))F ) ∈ N Q(A, B), which implies that (x1 ∗z1 )∗(y1 ∗z1 ) ∈ A, (x2 ∗z2 )∗(y2 ∗z2 ) ∈ A, (x3 ∗z3 )∗(y3 ∗z3 ) ∈ B and (x4 ∗ z4 ) ∗ (y4 ∗ z4 ) ∈ B. Since ỹ ∈ N Q(A, B), we have y1 , y2 ∈ A and y3 , y4 ∈ B. It follows from (2.14) that x1 , x2 ∈ A and x3 , x4 ∈ B. Hence x̃ = (x1 , x2 T, x3 I, x4 F ) ∈ N Q(A, B), and therefore N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S). Corollary 4.5. If A is a p-ideal of S, then N Q(A) is a neutrosophic quadruple p-ideal of N Q(S). Proposition 4.6. For any p-ideals A and B of S, the set N Q(A, B) satisfies the following implication. (∀x̃ ∈ N Q(S))(0̃ ⊙ (0̃ ⊙ x̃) ∈ N Q(A, B) ⇒ x̃ ∈ N Q(A, B)). (4.2) Proof. If 0̃ ⊙ (0̃ ⊙ x̃) ∈ N Q(A, B), then 0̃ ⊙ (0̃ ⊙ x̃) = (0, 0T, 0I, 0F ) ⊙ ((0, 0T, 0I, 0F ) ⊙ (x1 , x2 T, x3 I, x4 F )) = (0, 0T, 0I, 0F ) ⊙ ((0 ∗ x1 ), (0 ∗ x2 )T, (0 ∗ x3 )I, (0 ∗ x4 )F ) = (0 ∗ (0 ∗ x1 ), (0 ∗ (0 ∗ x2 ))T, (0 ∗ (0 ∗ x3 ))I, (0 ∗ (0 ∗ x4 ))F ) ∈ N Q(A, B). Hence (x1 ∗ x1 ) ∗ (0 ∗ x1 ) = 0 ∗ (0 ∗ x1 ) ∈ A, (x2 ∗ x2 ) ∗ (0 ∗ x2 ) = 0 ∗ (0 ∗ x2 ) ∈ A, (x3 ∗ x3 ) ∗ (0 ∗ x3 ) = 0 ∗ (0 ∗ x3 ) ∈ B and (x4 ∗ x4 ) ∗ (0 ∗ x4 ) = 0 ∗ (0 ∗ x4 ) ∈ B. Since A and B are p-ideals of S, it follows from (2.14) that x1 , x2 ∈ A and x3 , x4 ∈ B. Hence x̃ = (x1 , x2 T, x3 I, x4 F ) ∈ N Q(A, B). Corollary 4.7. For any p-ideal A of S, the set N Q(A) satisfies the following implication. (∀x̃ ∈ N Q(S))(0̃ ⊙ (0̃ ⊙ x̃) ∈ N Q(A) ⇒ x̃ ∈ N Q(A)). (4.3) Theorem 4.8. Let A and B be ideals of S. Then N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S) if and only if the following assertion is valid. (x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃) ∈ N Q(A, B) ⇒ x̃ ⊙ ỹ ∈ N Q(A, B) for all x̃, ỹ, z̃ ∈ N Q(S). (4.4) p-SEMISIMPLE NEUTROSOPHIC QUADRUPLE BCI-ALGEBRAS 35 Proof. Assume that N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S). Let x̃, ỹ, z̃ ∈ N Q(S) be such that (x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃) ∈ N Q(A, B). Then ((x̃ ⊙ ỹ) ⊙ (x̃ ⊙ ỹ)) ⊙ (((x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃)) ⊙ (x̃ ⊙ ỹ)) = 0̃ ⊙ (((x̃ ⊙ z̃) ⊙ (x̃ ⊙ ỹ)) ⊙ (ỹ ⊙ z̃)) = 0̃ ⊙ 0̃ = 0̃ ∈ N Q(A, B), and so x̃⊙ ỹ ∈ N Q(A, B) since N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S). Conversely, let A and B be ideals of S such that the set N Q(A, B) satisfies the condition (4.4). Then N Q(A, B) is a neutrosophic quadruple ideal of N Q(S) by Lemma 3.6, and so 0̃ ∈ N Q(A, B). Let x̃, ỹ, z̃ ∈ N Q(S) be such that (x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃) ∈ N Q(A, B) and ỹ ∈ N Q(A, B). Then x̃ ⊙ ỹ ∈ N Q(A, B) by (4.4), and thus x̃ ∈ N Q(A, B). Therefore N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S). Corollary 4.9. Given an ideal A of S, the set N Q(A) is a neutrosophic quadruple p-ideal of N Q(S) if and only if the following assertion is valid. (x̃ ⊙ z̃) ⊙ (ỹ ⊙ z̃) ∈ N Q(A) ⇒ x̃ ⊙ ỹ ∈ N Q(A) (4.5) for all x̃, ỹ, z̃ ∈ N Q(S). Theorem 4.10. Let A and B be ideals of S such that (∀x ∈ S)(0 ∗ (0 ∗ x) ∈ A (resp., B) ⇒ x ∈ A (resp., B)). (4.6) Then N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S). Proof. Let x, y, z ∈ S be such that (x ∗ z) ∗ (y ∗ z) ∈ A (resp., B) and y ∈ A (resp., B). Then (0 ∗ (0 ∗ ((x ∗ z) ∗ (y ∗ z)))) ∗ ((x ∗ z) ∗ (y ∗ z)) = (0 ∗ ((x ∗ z) ∗ (y ∗ z))) ∗ (0 ∗ ((x ∗ z) ∗ (y ∗ z))) = 0 ∈ A (resp., B). and so 0 ∗ (0 ∗ ((x ∗ z) ∗ (y ∗ z))) ∈ A (resp., B) since A and B are ideals of S. Now we have 0 ∗ (0 ∗ (x ∗ y)) = (0 ∗ y) ∗ (0 ∗ x) = (((0 ∗ z) ∗ (0 ∗ z)) ∗ y) ∗ (0 ∗ x) = (((0 ∗ (0 ∗ z)) ∗ z) ∗ y) ∗ (0 ∗ x) = (((0 ∗ y) ∗ (0 ∗ z)) ∗ z) ∗ (0 ∗ x) = ((0 ∗ (y ∗ z)) ∗ z) ∗ (0 ∗ x) = ((0 ∗ z) ∗ (0 ∗ x)) ∗ (y ∗ z) = ((0 ∗ (0 ∗ (0 ∗ z))) ∗ (0 ∗ x)) ∗ (y ∗ z) = ((0 ∗ (0 ∗ x)) ∗ (0 ∗ (0 ∗ z))) ∗ (y ∗ z) = (0 ∗ ((0 ∗ x) ∗ (0 ∗ z))) ∗ (y ∗ z) = (0 ∗ (0 ∗ (x ∗ z))) ∗ (y ∗ z) = (0 ∗ (y ∗ z)) ∗ (0 ∗ (x ∗ z)) = (0 ∗ (0 ∗ (0 ∗ (y ∗ z)))) ∗ (0 ∗ (x ∗ z)) = (0 ∗ (0 ∗ (x ∗ z))) ∗ (0 ∗ (0 ∗ (y ∗ z))) = 0 ∗ ((0 ∗ (x ∗ z)) ∗ (0 ∗ (y ∗ z))) = 0 ∗ (0 ∗ ((x ∗ z) ∗ (y ∗ z))) ∈ A (resp., B). It follows from (4.6) that x ∗ y ∈ A (resp., B). Hence x ∈ A (resp., B). This shows that A and B are p-ideals of S. Therefore N Q(A, B) is a neutrosophic quadruple p-ideal of N Q(S) by Theorem 4.4. Corollary 4.11. Let A be an ideal of S such that (∀x ∈ S)(0 ∗ (0 ∗ x) ∈ A ⇒ x ∈ A). (4.7) 36 G. MUHIUDDIN AND Y. B. JUN Then N Q(A) is a neutrosophic quadruple p-ideal of N Q(S). 5. C ONCLUSION In this paper, we consider characterizations of neutrosophic quadruple BCI-algebra, and give conditions for the neutrosophic quadruple BCI-set to be a p-semisimple BCIalgebra. Futhermore, we provide a condition for a subalgebra to be an ideal in neutrosophic quadruple BCI-algebra, and provide conditions for the set N Q(A, B) to be a neutrosophic quadruple closed ideal and neutrosophic quadruple p-ideal. We hope that this work will provide a deep impact on the upcoming research in this field and other related areas to open up new horizons of interest and innovations. Indeed, this work may serve as a foundation for further study of neutrosophic subalgebras in BCK/BCI-algebras. To extend these results, one can further study the neutrosophic set theory of different algebras such as MTL-algerbas, BL-algebras, MV-algebras, EQ-algebras, R0-algebras and Q-algebras etc. One may also apply this concept to study some applications in many fields like decision making, knowledge base systems, medical diagnosis, data analysis and graph theory etc. 6. ACKNOWLEDGEMENTS The authors are very thankful to the reviewers for careful detailed reading and helpful comments/suggestions that improved the overall presentation of this paper. R EFERENCES [1] A.A.A. Agboola, B. Davvaz and F. Smarandache. Neutrosophic quadruple algebraic hyperstructures, Ann Fuzzy math. Inform., 14 (1) (2017), 29–42. [2] S.A. Akinleye, F. Smarandache and A.A.A. Agboola. On neutrosophic quadruple algebraic structures, Neutrosophic Sets and Systems, 12 (2016), 122–126. [3] A. Borumand Saeid and Y.B. Jun. Neutrosophic subalgebras of BCK/BCI-algebras based on neutrosophic points, Ann. Fuzzy Math. 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M UHIUDDIN D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF TABUK , TABUK 71491, S AUDI A RABIA Email address: chishtygm@gmail.com YOUNG BAE J UN D EPARTMENT OF M ATHEMATICS E DUCATION , G YEONGSANG NATIONAL U NIVERSITY, J INJU 660-701, KOREA Email address: skywine@gmail.com ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 1, Number 1 (2018), 38-47 ISSN: 2582-0818 c http://www.technoskypub.com AN OBJECT ORIENTED APPROACH TO THE APPLICATION OF INTUITIONISTIC FUZZY SETS IN COMPETENCY BASED TEST EVALUATION P. A. EJEGWA∗ AND I. C. ONYEKE A BSTRACT. The theory of intuitionistic fuzzy sets (IFS) is a viable tool in decision science, robotics and control, medical imaging, among others. From the applications of IFS, it is proven that the concept of IFS is useful in providing a reliable and efficient framework or model to tackle uncertainty and vagueness embedded in decision making. In this paper, we explore the resourcefulness of IFS in competency based test evaluation (CBTE) for course selection into higher institution. We employ distance and similarity measures for IFS to achieve the test through a BESPOKE program developed using an object oriented programming language, JAVA to be specific. Using the output of the program in terms of distance and similarity between applicants and courses in intuitionistic fuzzy sense, we determine applicants suitable courses. 1. I NTRODUCTION The notion of intuitionistic fuzzy sets (IFS) proposed and studied in [1, 2, 3, 4, 6, 7] is a generalization of fuzzy sets with an additional degree of freedom called non-membership degree, when compared to fuzzy sets [26], which are fully described by the degree of membership only. In an IFS, the value of membership plus the value of non-membership for an element does not necessarily make one because of the possibility of hesitation. The additional degree of freedom means inherent possibility to model and process more adequately and more human consistently imprecise information, and makes the concept of IFS a useful tool in decision making [24]. The ability of expressing imprecise information leads to a construction of more reliable models. The use of these models is connected with processing of imprecise information via different measures. The measures of distance and similarity are the basic tools in applying IFS to decision making. See [16, 18, 23, 24] for details on distance measures and similarity measures, respectively. The concept of IFS seems to be a comprehensive tool for handling many aspects of imprecise information and as such, attracts much attention due to its significant in tackling 2010 Mathematics Subject Classification. 20N20, 03E72, 47S40. Key words and phrases. Competency based test evaluation; Distance measure; Fuzzy sets; Intuitionistic fuzzy sets; Object oriented programming language; Similarity measure. *Corresponding author. 38 AN OBJECT ORIENTED APPROACH TO THE APPLICATION OF INTUITIONISTIC FUZZY SETS ... 39 vagueness or the representation of imperfect knowledge in decision making. Many applications of IFS have been proposed and researched since inception in areas of medical diagnosis, medical imaging, career determination, appointment procedure, pattern recognition, supplier evaluation, etc. as seen in [5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 25]. In this paper, we explicate an application of IFS in CBTE for the purpose of course selection in a situation where the number of applicants is more than the available slots using the most accurate measure of the measures discussed in [23, 24] through a BESPOKE program developed using an object oriented programming language (JAVA). The output of the program in terms of distance and similarity between applicants and courses, determines a suitable course selections. The paper is organized thus. In Section 2, we recall some basic concepts of IFS while Section 3 discusses some selected distance and similarity measures between IFS together with a reliability analysis of the measures. In Section 4, we explore the advantage of IFS in CBTE using a JAVA programming language that utilizes the most accurate distance and similarity measures for the sake of efficiency. 2. P RELIMINARIES We recall some basic notions of IFS (cf. [1, 2, 3, 4, 6, 26]). Definition 2.1. Let X be a nonempty set. A fuzzy set A of X is characterized by a membership function µA : X → [0, 1]. That is,  if x ∈ X  1, 0, if x ∈ /X µA (x) =  (0, 1) if x is partly in X Alternatively, a fuzzy set A of X is an object having the form or A = {hx, µA (x)i | x ∈ X} A = {h µA (x) i | x ∈ X}, x where the function µA (x) : X → [0, 1] defines the degree of membership of the element x ∈ X. Definition 2.2. Let a nonempty set X be fixed. An IFS A of X is an object having the form A = {hx, µA (x), νA (x)i | x ∈ X} or µA (x), νA (x) A = {h i | x ∈ X}, x where the functions µA (x) : X → [0, 1] and νA (x) : X → [0, 1] define the degree of membership and the degree of non-membership, respectively of the element x ∈ X to A, which is a subset of X, and for every x ∈ X, For each A in X, 0 ≤ µA (x) + νA (x) ≤ 1. πA (x) = 1 − µA (x) − νA (x) 40 P. A. EJEGWA AND I. C. ONYEKE is the intuitionistic fuzzy set index or hesitation margin of x in X. The hesitation margin πA (x) is the degree of non-determinacy of x ∈ X, to the set A and πA (x) ∈ [0, 1]. The hesitation margin is the function that expresses lack of knowledge of whether x ∈ X or x∈ / X. Thus, µA (x) + νA (x) + πA (x) = 1. We denote the set of all intuitionistic fuzzy sets over X as IF S(X). Example 2.3. Let X = {1, 2, 3} be a fixed universe of discourse and A = {h1, 0.6, 0.1i, h2, 0.8, 0.1i, h3, 0.5, 0.3i} be the intuitionistic fuzzy set of X. The hesitation margins of the elements 1, 2, 3 to A are πA (1) = 0.3, πA (2) = 0.1 and πA (3) = 0.2 Definition 2.4. Let A, B ∈ IF S(X). Then, the following operations hold. (i) Inclusion A ⊆ B ⇔ µA (x) ≤ µB (x) and νA (x) ≥ νB (x)∀x ∈ X. (ii) Equality A = B ⇔ µA (x) = µB (x) and νA (x) = νB (x)∀x ∈ X. (iii) Complement Ac = {hx, νA (x), µA (x)i | x ∈ X} (iv) Union A ∪ B = {hx, max[µA (x), µB (x)], min[νA (x), νB (x)]i | x ∈ X} (v) Intersection A ∩ B = {hx, min[µA (x), µB (x)], max[νA (x), νB (x)]i | x ∈ X} (vi) Addition A ⊕ B = {hx, µA (x) + µB (x) − µA (x)µB (x), νA (x)νB (x)i | x ∈ X} (vii) Multiplication A ⊗ B = {hx, µA (x)µB (x), νA (x) + νB (x) − νA (x)νB (x)i | x ∈ X} Example 2.5. Let X = {1, 2, 3} be a fixed universe of discourse. Then and A = {h1, 0.6, 0.1i, h2, 0.8, 0.1i, h3, 0.5, 0.3i} B = {h1, 0.8, 0.1i, h2, 0.4, 0.3i, h3, 0.75, 0.1i} be the intuitionistic fuzzy sets of X. Clearly, A * B and B * A. Also, A 6= B. Ac = {h1, 0.1, 0.6i, h2, 0.1, 0.8i, h3, 0.3, 0.5i}, and B c = {h1, 0.1, 0.8i, h2, 0.3, 0.4i, h3, 0.1, 0.75i}, A ∪ B = {h1, 0.8, 0.1i, h2, 0.8, 0.1i, h3, 0.75, 0.1i}, A ∩ B = {h1, 0.6, 0.1i, h2, 0.4, 0.3i, h3, 0.5, 0.3i}, A ⊕ B = {h1, 0.92, 0.01i, h2, 0.88, 0.03i, h3, 0.875, 0.03i} A ⊗ B = {h1, 0.48, 0.19i, h2, 0.32, 0.37i, h3, 0.375, 0.37i} AN OBJECT ORIENTED APPROACH TO THE APPLICATION OF INTUITIONISTIC FUZZY SETS ... 41 3. D ISTANCE AND SIMILARITY MEASURES BETWEEN IFS S In this section, we consider some distance and similarity measures between IFSs studied in [19, 20, 23, 24]. Distance measure is a term that describes the difference between intuitionistic fuzzy sets and can be considered as a dual concept of similarity measure. Definition 3.1. Let X be nonempty set and A, B, C ∈ IF S(X). The distance measure d between A and B is a function d : IF S × IF S → [0, 1] satisfies (i) 0 ≤ d(A, B) ≤ 1 (boundedness) (ii) d(A, B) = 0 iff A = B (separability) (iii) d(A, B) = d(B, A) (symmetric) (iv) d(A, C) + d(B, C) ≥ d(A, B) (triangle inequality) (v) if A ⊆ B ⊆ C, then d(A, C) ≥ d(A, B) and d(A, C) ≥ d(B, C). Definition 3.2. Let X be nonempty set and A, B, C ∈ IF S(X). The similarity measure s between A and B is a function s : IF S × IF S → [0, 1] satisfies (i) 0 ≤ s(A, B) ≤ 1 (boundedness) (ii) s(A, B) = 1 iff A = B (separability) (iii) s(A, B) = s(B, A) (symmetric) (iv) s(A, C) + s(B, C) ≥ s(A, B) (triangle inequality) (v) if A ⊆ B ⊆ C, then s(A, C) ≤ s(A, B) and s(A, C) ≤ s(B, C). Remark. It follows that (i) d = 1 − s (ii) d(A, B) = d(Ac , B c ) (iii) s(A, B) = s(Ac , B c ). We make use of some measures proposed in [20, 23, 24] between IFSs, which were partly based on the geometric interpretation of IFS, and have some good geometric properties. Let A = {hx, µA (xi ), νA (xi ), πA (xi )i | x ∈ X} and B = {hx, µB (xi ), νB (xi ), πB (xi )i | x ∈ X} be two IFS in X = {x1 , ..., xn }, for i = 1, ..., n. Then, the distance measures are: Hamming distance; dH (A, B) = + Euclidean distance; dE (A, B) = + 1 n Σ (| µA (xi ) − µB (xi ) | + | νA (xi ) − νB (xi ) | 2 i=1 | πA (xi ) − πB (xi ) |) 1 ( Σni=1 [(µA (xi ) − µB (xi ))2 + (νA (xi ) − νB (xi ))2 2 1 (πA (xi ) − πB (xi ))2 ]) 2 42 P. A. EJEGWA AND I. C. ONYEKE normalized Hamming distance; dn−H (A, B) = dn−H (A, B) = + dH (A, B) ⇒ n 1 n Σ (| µA (xi ) − µB (xi ) | + | νA (xi ) − νB (xi ) | 2n i=1 | πA (xi ) − πB (xi ) |) normalized Euclidean distance; dn−E (A, B) = dn−E (A, B) = + dE (A, B) √ ⇒ n 1 n Σ [(µA (xi ) − µB (xi ))2 + (νA (xi ) − νB (xi ))2 2n i=1 1 (πA (xi ) − πB (xi ))2 ]) 2 ( Similarly, the following are the similarity measures of the aforementioned distance measures: Hamming similarity; sH (A, B) = + Euclidean similarity; sE (A, B) = + 1 1 − Σni=1 (| µA (xi ) − µB (xi ) | + | νA (xi ) − νB (xi ) | 2 | πA (xi ) − πB (xi ) |) 1 1 − ( Σni=1 [(µA (xi ) − µB (xi ))2 + (νA (xi ) − νB (xi ))2 2 1 (πA (xi ) − πB (xi ))2 ]) 2 normalized Hamming similarity; 1 n sn−H (A, B) = 1 − Σ (| µA (xi ) − µB (xi ) | + | νA (xi ) − νB (xi ) | 2n i=1 + | πA (xi ) − πB (xi ) |) normalized Euclidean similarity; 1 sn−E (A, B) = 1 − ( Σni=1 [(µA (xi ) − µB (xi ))2 + (νA (xi ) − νB (xi ))2 2n 1 + (πA (xi ) − πB (xi ))2 ]) 2 Now, we verify each of the distance and similarity measures with the aid of an example to ascertain the most accurate. The distance measure with the smallest value shows an accurate distance. Also, the greatest value of the similarity measure indicates the most reliable similarity. Example 3.3. Let X = {x, y, z} be a fixed universe of discourse. Then and A = {hx, 0.6, 0.2, 0.2i, hy, 0.8, 0.1, 0.1i, hz, 0.5, 0.3, 0.2i} B = {hx, 0.8, 0.2, 0.0i, hy, 0.7, 0.2, 0.1i, hz, 0.9, 0.1, 0.0i} be the intuitionistic fuzzy sets of X. AN OBJECT ORIENTED APPROACH TO THE APPLICATION OF INTUITIONISTIC FUZZY SETS ... 43 Now, using the distance measures above for i = 1, 2, 3, we have 1 dH (A, B) = (|0.6 − 0.8| + |0.2 − 0.2| + |0.2 − 0.0| + |0.8 − 0.7| 2 + |0.1 − 0.2| + |0.1 − 0.1| + |0.5 − 0.9| + |0.3 − 0.1| + |0.2 − 0.0|) 1 = (1.4) 2 = 0.7000 dE (A, B) = + = = 1 ( [(0.6 − 0.8)2 + (0.2 − 0.2)2 + (0.2 − 0.0)2 + (0.8 − 0.7)2 2 (0.1 − 0.2)2 + (0.1 − 0.1)2 + (0.5 − 0.9)2 + (0.3 − 0.1)2 + (0.2 − 0.0)2 ])1/2 r 1 (0.34) 2 0.4123 dn−H (A, B) dn−E (A, B) 1 (1.4) 6 = 0.2333 r 1 = (0.34) 6 = 0.2380 = The normalized Hamming distance yields the smallest distance between A and B. Hence, the most accurate. Similarly, from the similarity measures above, we get sH (A, B) = 0.3000, sE (A, B) = 0.5877, sn−H (A, B) = 0.7667, sn−E (A, B) = 0.7620 The normalized Hamming similarity gives the greatest similarity between A and B. Thus, the most reliable. Since both normalized Hamming distance and normalized Hamming similarity are the most reliable and accurate, we use both as the computational measures embedded to the program owing to their accuracy. 4. I NTUITIONISTIC FUZZY SETS IN CBTE USING JAVA PROGRAMMING LANGUAGE Competency based test evaluation (CBTE) is a viable tool in course selection into higher institutions. This is essential because many applicants are vying to study some courses with a limited slots. Placing an applicant with a requisite academic qualification to study a course enhances academic success. Many factors such as academic qualification, interest, personality make-up, etc. are indispensable in course selection. However, academic qualification is the only measureable factor as seem across the globe. In this section, we propose the application of IFS in CBTE via object oriented approach to curb the embedded imprecisions in course selection. In carry out the application, we assume that a set of applicants sit for a competency based test free from malpratice to ascertain how suitable they are to study their proposed courses. We use IFS as tool since it incorporates the membership degree (i.e. the applicant score), the non-membership degree (i.e. the marks of the questions the student failed) and the hesitation degree (which is the mark allocated to the questions the student do not attempt). 44 P. A. EJEGWA AND I. C. ONYEKE 4.1. Case study. Let A = {A1 , A2 , A3 , A4 , A5 } be the set of applicants for course selections, C = {medicine, pharmacy, surgery, anatomy, physiology} be the set of courses the applicants are vying for, and S = {English Language, Mathematics, Biology, Physics, Chemistry, Health Science} be the set of subjects related to the set of courses. Table 1 is a quasi-real database for courses and the related subjects’ require performance for course selections over 100%. medicine pharmacy surgery anatomy physiology English (0.8,0.1,0.1) (0.9,0.1,0.0) (0.5,0.3,0.2) (0.7,0.2,0.1) (0.8,0.1,0.1) Maths (0.7,0.2,0.1) (0.8,0.1,0.1) (0.5,0.2,0.3) (0.5,0.4,0.1) (0.5,0.3,0.2) Biology (0.9,0.1,0.0) (0.8,0.2,0.0) (0.9,0.1,0.0) (0.8,0.1,0.1) (0.9,0.1,0.0) Physics (0.6,0.3,0.1) (0.5,0.2,0.3) (0.5,0.4,0.1) (0.6,0.3,0.1) (0.6,0.2,0.2) Chemistry (0.8,0.2,0.0) (0.7,0.2,0.1) (0.7,0.1,0.2) (0.8,0.2,0.0) (0.7,0.2,0.1) Health Sci (0.8, 0.1,0.1) (0.8, 0.2,0.0) Table 1 (0.8, 0.1,0.1) (0.9, 0.1,0.0) (0.8, 0.1,0.1) Each score is described by three entries comprise of membership value, non-membership value and hesitation margin value. The applicants A = {A1 , A2 , A3 , A4 , A5 } sat for a competency based test in the aforesaid subjects and obtained the following marks over 100% as shown in Table 2. A1 A2 A3 A4 A5 English (0.6,0.3,0.1) (0.5,0.3,0.2) (0.7,0.1,0.2) (0.6,0.4,0.0) (0.8,0.1,0.1) Maths (0.5,0.4,0.1) (0.6,0.2,0.2) (0.6,0.3,0.1) (0.8,0.1,0.1) (0.7,0.1,0.2) Biology (0.6,0.2,0.2) (0.5,0.3,0.2) (0.7,0.1,0.2) (0.6,0.1,0.3) (0.8,0.2,0.0) Physics (0.5,0.3,0.2) (0.4,0.5,0.1) (0.5,0.4,0.1) (0.6,0.3,0.1) (0.7,0.1,0.2) Chemistry (0.5,0.5,0.0) (0.7,0.2,0.1) (0.4,0.5,0.1) (0.5,0.3,0.2) (0.6,0.1,0.3) Health Sci (0.6,0.2,0.2) (0.7,0.1,0.2) Table 2 (0.6,0.3,0.1) (0.7,0.2,0.1) (0.8, 0.1,0.1) The scores in both Table 1 and Table 2 are marks obtained out of 100% in intuitionistic fuzzy values of the set S = {English Language, Mathematics, Biology, Physics, Chemistry, Health Science}. 4.2. Algorithm for calculating distance and similarity measures between applicants and courses. We present algorithm of normalized Hamming distance and normalized Hamming similarity. S = {English Language, Mathematics, Biology, Physics, Chemistry, Health Science} is the set of subjects under consideration. Recall dn−H (A, C) = 1 n Σi=1 (| µA (si ) − µC (si ) | + | νA (si ) − νC (si ) | + | πA (si ) − πC (si ) |) 2n and sn−H (A, C) = 2n − Σn i=1 (| µA (si ) − µC (si ) | + | νA (si ) − νC (si ) | + | πA (si ) − πC (si ) |) 2n are the normalized Hamming distance and normalized Hamming similarity measures between A = {A1 , A2 , A3 , A4 , A5 } and C = {medicine, pharmacy, surgery, anatomy, physiology}, where i = 1, ..., 6. 4.2.1. Algorithm for calculating normalized Hamming distance and normalized Hamming similarity between applicants and courses. Precondition: ac, cs, as are object references to a collection of applicant course, course subject and applicant subject entities. AN OBJECT ORIENTED APPROACH TO THE APPLICATION OF INTUITIONISTIC FUZZY SETS ... 45 Postcondition: the algorithm on steps 1-13 compute the normalized Hamming distance and normalized Hamming similarity, respectively. 1: retrieve a collection of applicant course record as ac and course subject record as cs 2: repeat steps 3-6 while (as! =null and as.isEmpty()) 3: repeat steps 4 and 5 while (cs! =null && !cs.isEmpty()) 4: for each as and cs compute normalized Hamming distance as 1 n Σi=1 (| µA (si ) − µC (si ) | + | νA (si ) − νC (si ) | + | πA (si ) − πC (si ) |) 2n 5: 6: 7: 8: 9: 10: persist applicant courses database with computed normalized Hamming distance end while end while repeat steps 9-12 while (as! =null and as.isEmpty()) repeat steps 10 and 11 while (cs! =null && !cs.isEmpty()) for each as and cs compute normalized Hamming similarity as 2n − Σn i=1 (| µA (si ) − µC (si ) | + | νA (si ) − νC (si ) | + | πA (si ) − πC (si ) |) 2n 11: 12: 13: 14: persist applicant courses database with computed normalized Hamming similarity end while end while exit 4.3. Results and decision. Using the algorithm above via JAVA programming language, the following results are obtained. The choice of JAVA is because of its portability, architecture neurality, robustness, security raptness and multi-threaded nature. A1 A2 A3 A4 A5 medicine 0.2167 0.2000 0.1833 0.1833 0.1333 pharmacy 0.2333 0.2333 0.2167 0.1833 0.1500 surgery 0.2167 0.1333 0.2000 0.2000 0.1667 Table 3 anatomy 0.1667 0.2000 0.1833 0.2000 0.2000 physiology 0.2000 0.2000 0.1833 0.2167 0.1333 A1 A2 A3 A4 A5 medicine 0.7833 0.8000 0.8167 0.8167 0.8667 pharmacy 0.7667 0.7667 0.7833 0.8167 0.8500 surgery 0.7833 0.8667 0.8000 0.8000 0.8333 Table 4 anatomy 0.8333 0.8000 0.8167 0.8000 0.8000 physiology 0.8000 0.8000 0.8167 0.7833 0.8667 Table 3 is gotten using steps 1-7 while Table 4 is gotten using steps 1, 8-13. In Table 3, the smallest value gives the CBTE of the applicants. Also, the greatest value provides the CBTE of the applicants in Table 4. From both Table 3 and Table 4, A1 is suitable for anatomy, A2 is suitable for surgery, A3 is suitable for medicine, anatomy and physiology, A4 is suitable for both medicine and pharmacy, and A5 is suitable for both medicine and physiology. We observe that A3 , A4 and A5 have the leeway to choice based on their personal interest from the courses they are suitable for. 46 P. A. EJEGWA AND I. C. ONYEKE 5. C ONCLUSIONS We have proposed an application of IFS in CBTE via object oriented approach embedded with normalized Hamming distance and normalized Hamming similarity. We conclude that IFS theory is a decisive tool use in critical decision making problems like this. It is observed that without IFS, this exercise would have been compromised with a consequent effect on the applicants. The object oriented approach disscussed in the work could be extended to other measures for easily application to multi-criteria decision making problems in future research. 6. ACKNOWLEDGEMENTS The authors are thankful to the Editor-in-Chief, Prof. G. Muhiuddin for the technical comments and to the anonymous reviewers for their relevant suggestions. R EFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] K. T. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, 1983. K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set Syst., 20 (1986), 87–96. K. T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Set Syst., 33 (1989), 37–46. K. T. Atanassov, New operations defined over intuitionistic fuzzy sets, Fuzzy Set Syst., 61 (1994), 137–142. K. T. Atanassov, Intuitionistic fuzzy sets: theory and applications, Physica-Verlag, Heidelberg, 1999. K. T. Atanassov, On intuitionistic fuzzy sets theory, Springer, Berlin, 2012. K. T. Atanassov, G. Cuvalcioglu and V. Atanassova, A new modal operator over intuitionistic fuzzy sets, Note IFS, 20(5) (2014), 1–8. T. 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C. O NYEKE D EPARTMENT OF M ATHEMATICS /S TATISTICS /C OMPUTER S CIENCE , U NIVERSITY OF AGRICULTURE , P.M.B. 2373, M AKURDI , N IGERIA Email address: onyeke.idoko@gmail.com ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 1, Number 1 (2018), 48-64 ISSN: 2582-0818 c http://www.technoskypub.com FUZZY SHORTEST PATH IN AN INTERVAL-VALUED FUZZY HYPERGRAPH USING SIMILARITY MEASURES TARASANKAR PRAMANIK∗ AND MADHUMANGAL PAL A BSTRACT. In this paper, a new approach is introduced to find the shortest path between two given vertices on interval-valued fuzzy hypergraphs. When only crisp numbers are not sufficient to measure a real world parameter, fuzzy numbers are considered. But, there are many types of fuzzy numbers available in literature. Most useful fuzzy number is trapezoidal fuzzy numbers. Throughout this paper the interval-valued trapezoidal fuzzy number is used as the arc length of the interval-valued fuzzy hypergraph. We have measured similarity between two interval-valued fuzzy numbers to find the shortest path. An algorithm is also designed to find all possible hyperpaths in a hypergraph and calculated its time complexity. 1. I NTRODUCTION The shortest path problem mainly traces on finding minimum distance between two specified vertices in a graph. The fuzzy shortest path problem was first analysed by Duboids and Prade [15] using edge weight as fuzzy number instead of a real number. Later, the concept of degree of possibility in which an arc is a shortest path is introduced by Okada [24]. Takahashi and Yamakami [43] discussed the shortest path problem from a specified node to all other nodes on a network. Chuang and Kung introduced several methods to solve this kind of problem. In [14], they proposed a procedure that can find a fuzzy shortest path among all possible paths in a network. There is another approach to find fuzzy shortest path, namely fuzzy linear programming approach, has been found by Lin and Chern [22]. Nayeem and Pal [23] have designed algorithms to solve fuzzy shortest path problem on a network with imprecise edge weights. Chuang and Kung [13] have introduced a new algorithm for discrete fuzzy shortest path problem in a network. Recently, Kumar et. al. [21] have worked on algorithms to find the shortest path in fuzzy graphs with interval-valued intuitionistic fuzzy edge weights. Akram and Davvaz [1] discussed strong intuitionistic fuzzy graphs. A novel application of intuitionistic fuzzy digraphs in decision support systems is given in [2]. Akram and Dudek [3] discussed intuitionistic fuzzy hypergraphs and provided an application. Also, Akram and Al-Shehrie defined intuitionistic fuzzy cycles and intuitionistic fuzzy tree [4], 2010 Mathematics Subject Classification. 05C85, 05C90, 68R05. Key words and phrases. fuzzy number; fuzzy hyperpaths; interval-valued fuzzy hypergraph; fuzzy shortest hyperpath; algorithm. *Corresponding author. 48 FUZZY SHORTEST PATH IN IVFHG 49 bipolar fuzzy competition graphs [5] and intuitionistic fuzzy planar graphs [6]. Sahoo and Pal [31] discussed the concept of intuitionistic fuzzy competition graph. They also discuss different types of products on intuitionistic fuzzy graphs [30], product of intuitionistic fuzzy graphs and degree [32], etc. Many researchers have focused on fuzzy shortest path problem in a network due to its importance to many applications such as communications, routing, transportation, etc. In traditional shortest path problems, the arc lengths of the network are taken as precise numbers, but in the real-world problem, the arc length may represent transportation time or cost which can be known only approximately due to vagueness of information, and hence it can be considered as a fuzzy number. Here, we considered a special type of fuzzy numbers namely, trapezoidal fuzzy number. A hypergraph is a generalization of a graph in which an edge can connect any number of vertices. Formally, a hypergraph H is a pair H = (X, E) where X is a set of elements called nodes or vertices and E is a set of non-empty subsets of X called hyperedges or edges. Therefore, E is a subset of P (X)\{φ}, where P (X) is the power set (collection of all subsets) of X. While edges of a graph are pair of nodes or vertices, hyperedges are arbitrary sets of nodes and therefore contain arbitrary number of nodes. A directed hypergraph is a generalization of the concept of directed graph. It was first introduced in [7] to represent functional dependencies in relational data base. A directed hypergraph is given by a set of nodes V and a set of pairs (T, h) (hyperedges) where T is a subset of V and h is a single node in V . The most obvious interpretation of a hyperedge (T, h) is that the information associated to h functionally depends on the information associated to nodes in T . A hypergraph is useful in various combinatorial structures that generalize graphs. Directed hypergraph is an extension of directed graphs, and have often used in several areas such as a modelling and algorithmic tool. A brief introduction to directed hypergraphs is given by Gallo et al. [17]. Geotschel [18] introduced the concept of fuzzy hypergraphs and Hebbian structures. Goetschel [19] also explained the coloring of fuzzy hypergraphs. Intersecting fuzzy hypergraphs are defined by Goetschel [20]. Samanta and Pal have also worked on bipolar fuzzy hypergraphs and related fuzzy graphs in [26–28, 33–42]. Readers can found many recent works on [25, 29]. Interval valued fuzzy set is a generalization of traditional fuzzy set. So it is more adequate to describe the uncertainty than the traditional fuzzy sets. It is therefore important to use interval valued fuzzy sets in applications such as fuzzy control, network topology, transportation, etc. When in a fuzzy graph the arc (edge) weights and/or vertex weights are considered as the interval-valued fuzzy sets, the resultant graph becomes an interval-valued fuzzy graph. There are many real life problems such as communications, routing, transportation, etc., finding shortest path is very essential in research purposes. But finding shortest paths in hypergraphs are too more demanding in recent research areas. Hypergraphs can consider more complex networks such as protein-protein interaction network, social networks, information theory, publication data, collaborations, chemical processes, etc. Hypergraphs are learned to segment or classify the datas. In learning of distances between two nodes in a hypergraph, first create a normal graph by connecting nodes with weighted edges. Weight is the sum of the weights of hyperedges traversed in the shortest path. An example is given here. Suppose there are seven cities, say, A, B, C, D, E, F , G in a country. The hypergraph relation of these cities are given in Figure 1. We can construct the normal graph 50 TARASANKAR PRAMANIK AND MADHUMANGAL PAL F IGURE 1. Hypergraph of seven cities by specifying connections to each cities which are connected by roads such as A −→ B, A −→ C, A −→ D, B −→ E, C −→ E, C −→ F , D −→ F , E −→ G, F −→ G. The corresponding graph is shown in Figure 2. F IGURE 2. Graph of Figure 1 The population of G has demand of some products. The city A has a supplier to supply the products but wants the minimum cost of transportation to supply. Each road has cost of transportaion. Depending on many parameters (such as toll taxes, levies, security taxes, etc.) costs of transportation varies road by road and also time by time. So the costs can be treated as interval-valued trapezoidal fuzzy numbers where left end trapezoidal fuzzy number is the minimum cost and right end trapezoidal fuzzy number is the maximum cost to transport by that road. So this problem can be modelled as the interval-valued fuzzy hypernetwork with edge weights as trapezoidal fuzzy numbers. Being motivated from this essence, an algorithm is designed to find shortest path of interval-valued fuzzy hypernetwork. Though there are many research articles to find shortest paths, here we have generalized the fuzziness by introducing interval-valued fuzzy numbers and trapezoidal fuzzy numbers together. The interval-valued trapezoidal fuzzy numbers are the generalization of interval-valued fuzzy numbers, trapezoidal fuzzy numbers and also fuzzy numbers. All these numbers can be obtained by considering particular cases. In this paper, we have designed an algorithm to find the fuzzy shortest path on a fuzzy hypernetwork. The membership values of the edges are taken as interval-valued trapezoidal fuzzy number. The remaining part of the paper is organized as follows: FUZZY SHORTEST PATH IN IVFHG 51 In Section 2, preliminaries of the main work is introduced. In Section 3, an algorithm based on BFS (Breadth-First Search) technique is described to find all the paths between two nodes. Section 4 computes the fuzzy shortest hyperpaths of a network. At the end of the paper, conclusion has been drawn. 2. P RELIMINARIES In this section, the definition of interval-valued fuzzy hypergraph, directed intervalvalued fuzzy hypergraph, interval-valued trapezoidal fuzzy number, similarity measures of two interval-valued trapezoidal fuzzy numbers are given. These are the basic concepts required to design the algorithm to find the fuzzy shortest hyperpath. A fuzzy set F on a universal set X is defined by a mapping m : X → [0, 1], which is called the membership function. A fuzzy set is denoted by F = (X, m). A fuzzy graph G = (V, σ, µ) is a non-empty set V together with a pair of functions σ : V → [0, 1] and µ : V × V → [0, 1] such that for all x, y ∈ V, µ(x, y) ≤ min{σ(x), σ(y)}, where σ(x) represents the membership values of the vertex x and µ(x, y) represents membership values of the edge (x, y) in G. A loop at a vertex x in a fuzzy graph is represented by µ(x, x) 6= 0. An edge is non-trivial if µ(x, y) 6= 0. A fuzzy graph G = (V, σ, µ) is complete if µ(x, y) = min{σ(x), σ(y)} for all x, y ∈ V , where (x, y) denotes the edge between the vertices x and y. An interval-valued fuzzy set A on a set X is defined by A = {(x, [σ L (x), σ U (x)] : x ∈ X} where the membership functions σ L (x), σ U (x) are such that σ L (x) ≤ σ U (x) and σ L (x), σ U (x) ∈ [0, 1] for all x ∈ X. An interval-valued fuzzy graph ξ is denoted by ξ = (V, A, B), where A = (V, [σ L , σ U ]) is an interval-valued fuzzy set on V and B = (V × V, [µL , µU ]) is an interval-valued fuzzy set on V ×V , such that µL (x, y) ≤ min{σ L (x), σ L (y)} and µU (x, y) ≤ min{σ U (x), σ U (y)} for all x, y ∈ V . We call A as the interval-valued fuzzy vertex set of ξ and B as the intervalvalued fuzzy edge set of ξ respectively. A hypergraph is a generalization of a graph in which an edge can connect any number of vertices. Formal definition is as follows: Definition 2.1 (Hypergraph). Let X = {x1 , x2 , . . . , xn } be a finite set. A hypergraph on X is a family H = {E1 , E2 , . . . , Em } of subsets of X such that (i) Ei 6= φ, (i = 1, 2, . . . , m) and m [ Ei = X. (ii) i=1 The elements x1 , x2 , . . . , xn are called the vertices and the sets E1 , E2 , . . . , Em are called the hyperedges (or, simply edges) of the hypergraph. A simple hypergraph is a hypergraph H = {E1 , E2 , . . . , Em } such that Ei ⊂ Ej ⇒ i = j. A graph is a simple hypergraph each of whose edges has cardinality 2. LetSX be a finite set and let E be a family of non-empty fuzzy subsets of X such that X = {supp A|A ∈ E}. Then the pair H = (X, E) is called a fuzzy hypergraph on X. Let X be a finite set S and let E be a family of non-empty interval-valued fuzzy subsets of X such that X = {supp A|A ∈ E}. Then the pair H = (X, E) is called an intervalvalued fuzzy hypergraph on X. In classical graph theory, parameters are measured by a single number e.g., 5 km (for distances), 30 kg (for weights), etc. But in practical situation, a single number could not give enough information of the parameters. It may come in an imprecise way like ‘about 5 52 TARASANKAR PRAMANIK AND MADHUMANGAL PAL km’, ‘between 10-15 yards’, etc. In this paper, the arc lengths of a hypergraph are taken as interval number. In general, an interval number is defined as I = [aL , aR ] = {a : aL ≤ a ≤ aR } where aL and aR are the real numbers called the left end point and the right end point of the interval I respectively. Besides interval number there are so many fuzzy numbers viz. triangular fuzzy number, trapezoidal fuzzy number, etc. e = [a1 , a2 , a3 , a4 ; w e], a1 ≤ a2 ≤ a3 ≤ a4 and 0 ≤ A trapezoidal fuzzy number A A wAe ≤ 1 is represented by the membership function (shown in Figure 3) as  0, x ≤ a1    x−a1  w ,  a2 −a1 Ae a1 < x ≤ a2 wAe, a2 < x ≤ a3 µAe(x) =  −x  w , a3 < x ≤ a4  aa44−a e A  3  0, x ≥ a4 . e = [a1 , a2 , a3 , a4 ; w e], Chen [8, 9] defines the generalized trapezoidal fuzzy number A A a1 ≤ a2 ≤ a3 ≤ a4 and 0 ≤ wAe ≤ 1 where, wAe represents the degree of confidence of the liguistic opinion. Chen and Chen [10] introduces the Simple Center of Gravity ∗ Method (SCGM) to calculate the Center of Gravity (COG) point (x∗Ae, yA e) of a generalized trapezoidal fuzzy number as ( a −a wAe × a3 −a2 +2 4 1 ∗ , if a1 6= a4 yA 6 e = wAe , if a1 = a4 . 2 x∗Ae = ∗ ∗ yA e − yA e(a3 + a2 ) + (a4 + a1 )(wA e) 2wAe . ✻ 1 wAe ✲ a1 a2 a3 a4 F IGURE 3. Trapezoidal fuzzy number ≈ eL , A eU ] as Yao and Lin [45] studied the interval-valued trapezoidal fuzzy number A = [A eL = (aL , aL , aL , aL ; w eL ) and A eU = (aU , aU , aU , aU ; w eU ) shown in Figure 4, where each A 1 2 3 4 1 2 3 4 A A L U e e are trapezoidal fuzzy numbers and A ⊆ A . FUZZY SHORTEST PATH IN IVFHG 53 Definition 2.2 (Interval-valued trapezoidal fuzzy number). An interval-valued trapezoidal ≈ ≈ U L U fuzzy number (IVTFN) A is denoted by A = {(µL e(x), µA e (x))|µA e(x) ≤ µA e (x), x ∈ X}, A where  0,     x−aL 1    aL2 −aL1 wAeL , L wAeL , µAe(x) =   aL 4 −x   L −aL wA eL ,  a   4 3 0, x ≤ aL 1 L aL 1 < x ≤ a2 L L a2 < x ≤ a3 L aL 3 < x ≤ a4 x ≥ aL 4, µ ≈ (x)  0,     x−aU 1    aU2 −aU1 wAeU , U wAeU , and µAe (x) =   aU 4 −x   U −aU wA eU ,  a   4 3 0, x ≤ aU 1 U aU 1 < x ≤ a2 U U a 2 < x ≤ a3 U aU 3 < x ≤ a4 x ≥ aU 4. A 1 wAeU ❊ wAeL aU 1 U L aL 1 a2 a2 ❊ ❊ ❊ ❊ ❊ L U aL 3 a4 a3 aU 4 X F IGURE 4. Interval-valued trapezoidal fuzzy number L U L U U L U L U Since µL A (x) ≤ µA (x), therefore a1 ≥ a1 , a2 ≥ a2 and a3 ≤ a3 , a4 ≤ a4 must hold. Interval-valued fuzzy numbers is used by Lin [16] to represent vague processing time in job-shop scheduling problems. Wang and Li [44] present the correlation coefficient of interval valued fuzzy numbers and some of their properties. Yao and Lin used intervalvalued fuzzy numbers to represent unknown job processing time for constructing a fuzzy flow-shop sequencing model. Some methods have been proposed in [10,11] for measuring the degree of similarity between interval-valued fuzzy numbers. In [12], Chen et al. represents a method to measure the similarity between two IVTFNs. In this paper, we used the method to compare two IVTFNs. ≈ ≈ ≈ eL , A eU ] = [(aL , aL , aL , aL ; w eL ), Consider two IVTFNs A and B, where A = [A ≈ 1 2 3 4 A U U U L L L L U U U U eL eU (aU e L ), (b1 , b2 , b3 , b4 ; eU )] and B = [B , B ] = [(b1 , b2 , b3 , b4 ; wB 1 , a 2 , a 3 , a4 ; w A ≈ ≈ wBeU )]. The degree of similarity [12] between IVTFNs is denoted by S(A, B) and is defined by ≈ ≈ S(A, B) = eU , B e U ) × (1 + S(A e∆ , B e ∆ )) S(A 2 54 TARASANKAR PRAMANIK AND MADHUMANGAL PAL e∆ , B e ∆ ) = degree of similarity between the distance values of the two IVTFNs where, S(A  = 1 − qP L U L ∆ai = |aU i − ai | and ∆bi = |bi − bi |  " # r 4 2 i=1 (∆ai − ∆bi )  × 1 − |∆Sa − ∆Sb | 2 2 × 1− |wAeL − wBeL | × T ∆. |wAeU − wBeU | Here ∆Sa = |SAeU −SAeL | and ∆Sb = |SBeU −SBeL |. T ∆ denotes map distance between ≈ eL and A eU of IVTFN A. The parameters the lower and upper trapezoidal fuzzy numbers A ∆ SAeL , SAeU , SBeL , SBeU and T can be calculated as follows: s s P4 P4 L − āL )2 L L 2 (a i=1 i i=1 (bi − b̄ ) , SBeL = SAeL = n−1 n−1 s s P4 P 4 U U U 2 U 2 i=1 (ai − ā ) i=1 (bi − b̄ ) SAeU = , SBeU = n−1 n−1 i h 1+max{|∆a4 −∆a3 |,|∆b4 −∆b3 |} 2 −∆a1 |,|∆b2 −∆b1 |} + 2 − 2 − 1+max{|∆a 1+min{|∆a2 −∆a1 |,|∆b2 −∆b1 |} 1+min{|∆a4 −∆a3 |,|∆b4 −∆b3 |} and T ∆ = 2 U U U where āU denotes the average of the four values aU 1 , a2 , a3 , a4 at the upper trapezoidal U e fuzzy number A and the similar concept is used for the notations āL , b̄U and b̄L and qP  "  # r 4 U − bU ) 2 (a min{wAeU , wBeU } |S − S | i i=1 i U U e e U U A B e ,B e ) = 1 − × 1− × × TU S(A 2 2 max{wAeU , wBeU } h i 1+max{|aU −aU |,|bU −bU |} 1+max{|aU −aU |,|bU −bU |} 2 − 1+min{|aU2 −aU1 |,|bU2 −bU1 |} + 2 − 1+min{|aU4 −aU3 |,|bU4 −bU3 |} 2 1 2 1 4 3 4 3 where T U = . 2 ≈ ≈ ≈ The larger the value of S(A, B), the greater the similarity between the IVTFNs A and ≈ B. Next, we consider the addition of two IVTFNs. ≈ eL , A eU ] = [(aL , aL , aL , aL ; w eL ), Definition 2.3 (Addition of two IVTFNs). Let A = [A 1 2 3 4 A ≈ L L L L U U U U U U U eL eU (aU eU )] and B = [B , B ] = [(b1 , b2 , b3 , b4 ; wB e L ), (b1 , b2 , b3 , b4 ; 1 , a2 , a 3 , a4 ; w A ≈ ≈ wBeU )] be two IVTFNs then the addition of these IVTFNs is denoted by A + B and is ≈ ≈ U U L L L L L L L defined by A + B = [(aL eL , wB e L }), (a1 + b1 , 1 + b1 , a2 + b2 , a3 + b3 , a4 + b4 ; max{wA U U U U U aU eU , wB e U })]. 2 + b2 , a3 + b3 , a4 + b4 ; max{wA In general, networks are directed graphs. So directed interval-valued fuzzy hypergraphs are considered. In a directed network, two nodes may connect with two different edges each with opposite direction but in a simple undirected network, two nodes are connected by only one edge. Now, we define following terms: Definition 2.4 (Directed interval-valued fuzzy hypergraph). A directed interval-valued → − → − fuzzy hypergraph G is a pair (V, E ) where V is a non-empty set of vertices (called nodes) → − and E is the set of interval-valued fuzzy hyperarcs; an interval-valued fuzzy hyperarc FUZZY SHORTEST PATH IN IVFHG 55 → − e ∈ E is defined as a pair (T (e), h(e)), where T (e) is a subset of V , with T (e) 6= φ is its → − tail and h(e) is a vertex in V − T (e) is its head. A node s is a source node in G if h(e) 6= s → − → − for every e ∈ E and a node d is said to be a destination node if d ∈ / T (e) for every e ∈ E . → − Definition 2.5 (Valid ordering in a fuzzy hypergraph). Let G = (V, E ) be a directed fuzzy hypergraph. A valid ordering in G is a lexicographic ordering of nodes V = {v1 , v2 , · · · , vn }, such that for any e ∈ E (vi ∈ T (e)) and (h(e) = vj ) ⇒ i < j. For example, the hypergraph shown in Figure 5 has a valid ordering of nodes namely, {1, 2, 3, 4, 5, 6, 7, 8}. e1 e2 1♠ ✒ 2♠ ✶ 3♠ e5 ③ e6 e3 e4 q 4♠ ✿ e7 q ♠ 6❍ ❍❍ e 8 ❍ ✲❍ ❥ 8♠ ❍ ❥ ✲ 7♠ ⑦ 5♠ F IGURE 5. An example of hypernetwork Definition 2.6 (Minimal fuzzy hypergraph). A minimal fuzzy hypergraph is a fuzzy hypergraph in which after deletion of a fuzzy vertex (node) or fuzzy hyperarc, the resultant graph is not a fuzzy hypergraph. → − Definition 2.7 (Fuzzy hyperpath). Consider a directed fuzzy hypergraph G = (V, E ). A fuzzy hyperpath Πst of source s and destination t is a minimal fuzzy hypergraph Gπ = (Vπ , Eπ ) with valid ordering of nodes satisfying the following conditions: [ (T (e) ∪ {h(e)}), (1) Eπ ⊆ E and Vπ = e∈Eπ (2) s, t ∈ Vπ , (3) u ∈ Vπ \{s} ⇒ u is connected to s in Gπ . For example, in Figure 5, 1 → 3 → 7 → 8 is a fuzzy hyperpath. 3. E NUMERATION OF ALL HYPERPATHS IN A HYPERGRAPH In this section, we describe an algorithm based on BFS technique to find the paths between two nodes. For the sake of algorithm we use some notations. Let Pq be a queue of edges forming the path and Vq is a queue of vertices. enqueue(v) adds the vertex v to the queue Vq and dequeue removes a vertex from the queue Vq . In this algorithm, we use the following functions: i) Array(v): array of vertices associated to a vertex v. 56 TARASANKAR PRAMANIK AND MADHUMANGAL PAL ii) Arrays(v): array of elements associated to a vertex v; an element of Arrays(v) is either an array or a single vertex. iii) CreateArray(arrays, vertex to add): creates a new array by adding a vertex vertex to add to the existing array arrays. iv) AddArray(arrays, vertex to add): adds a vertex vertex to add to the associated array arrays. Initially, every vertex is labeled as ‘not visited’. After each iteration when a vertex is picked up and assigned to an array the vertex is immediately being labeled as ‘visited’. In this algorithm vertex.visited denotes the label of a vertex as either ‘visited’ or ‘not visited’. Algorithm PF Input:: Source node s and the destination node t of the directed fuzzy hypergraph → − G = (V, E ). Output:: All hyperpaths. Step 1.: Create an array with one element s and set it to the vertex s. Each time we call this array by Array(s), the associated array for the vertex s. If a vertex has two or more arrays then all the arrays are stored in a single array called Arrays(s). → − Step 2.: For each e ∈ E if s ∈ T (e) then Vq ← enqueue(h(e)) if h(e).visited =‘visited’ then CreateArray(Arrays(s), h(e)) \\ Create a new array for the vertex s\\. else AddArray(Arrays(s), h(e)) \\ Add the vertex h(e) to Arrays(s)\\. end if; Set h(e).visited =‘visited’. end if; end for; Step 3.: s ← dequeue(Vq ). Step 4.: If Vq is not empty, then go to Step 2 otherwise Stop the process. end PF. At the end of the algorithm an array is created for each vertex. We find the arrays of the destination vertex t from Arrays(t). All the arrays Array(t) of Arrays(t) are all the possible hyperpaths. 3.1. Proof of correctness of the algorithm. In Algorithm PF, every vertex v is associated with an array Arrays(v), where Arrays(v) contain some array of vertices Array(v1 ), Array(v2 ), . . . , Array(vk ). Each of these arrays determines path from source node to the vertex v. We claim that Algorithm PF determines all the hyperpaths between two given nodes. First, Algorithm PF sets an array Arrays(v) with only one element s which is the source node. In a hypergraph all the paths are of the form s ∈ T (e1 ) → h(e1 ) → T (e2 ) → h(e2 ) → · · · → T (em ) → h(em ) = t, where t is the destination node. Step 2 determines all these paths by checking each e if s ∈ T (e). s ∈ T (e) implies that there exist a path from s to h(e) along the edge e. This h(e) is the next starter of the edge connected to s. This algorithm stores the path traversed FUZZY SHORTEST PATH IN IVFHG 57 from s to h(e). In this way, all the paths are traversed and found the path. Since each path from the source vertex to a vertex is stored along with that vertex then Arrays(t) of the destination vertex t has all the paths from source node to destination node. Theorem 3.1. The Algorithm PF runs in O(mn) time, where m is the number of edges and n is the number of vertices. Proof. Let the processor takes unit time to perform a single instruction. Step 1 of the Algorithm PF takes O(1) time. The algorithm consists of a loop from Step 2 to Step 4. This loop carry over O(n) times as Vq contains only the vertices of the graph. Within this loop we see that a loop occurs in Step 2 which is terminated after m times. Hence the overall time complexity of the Algorithm PF is of O(mn). Illustration of algorithm PF. Consider a directed hypergraph (hypernetwork) shown in Figure 5. Step 1: Consider the source vertex as 1 and the destination vertex as 8. Assign an array (1) with only one entry 1 to the vertex 1. Step 2: 1 ∈ T (e1 ), 1 ∈ T (e2 ), 1 ∈ T (e3 ) and 1 ∈ T (e4 ). Step 3: h(e1 ) = 2. So the vertex 2 is enqueued to the queue Vq . Step 4: The vertex 2 is marked as visited and we create an array (1, 2). Step 5: Similarly, the vertices 3 and 4 are marked as visited and queued to Vq and we set the arrays (1, 3) and (1, 4) associated to 3 and 4 respectively. Step 6: Dequeue the vertex 2 from the queue Vq and see that 2 ∈ T (e5 ). Then h(e5 ) = 6. Step 7: The vertex 6 is marked as visited and queued to Vq . Assign an array (1, 2, 6) to the vertex 6. Step 8: For the vertex 3 of Vq , we see that h(e5 ) = 6 is visited then we create a new array (1, 3, 6) and assigned to 6. Step 9: Proceeding in the similar way we assign arrays to each vertices. And observe that the destination vertex 8 is assigned with the arrays (1, 2, 6, 8), (1, 3, 6, 8), (1, 3, 7, 8), (1, 4, 7, 8), (1, 5, 7, 8). Therefore all the possible hyperpaths in the given hypernetwork are P-1: 1 → 2 → 6 → 8, P-2: 1 → 3 → 6 → 8, P-3: 1 → 3 → 7 → 8, P-4: 1 → 4 → 7 → 8, P-5: 1 → 5 → 7 → 8. The length of a hyperpath of a network is the sum of the lengths of all hyperarcs on the hyperpath. Here, we describe an algorithm to find the minimum length of all fuzzy hyperpaths of the interval-valued fuzzy hypergraph from a given source node to a destination node. Algorithm MLIVFH : \\This algorithm determines the minimum weighted (trapezoidal fuzzy) hyperpaths of an interval-valued fuzzy hypergraph between two given nodes.\\ ≈ Input: Weights (Ai ) (i = 1, 2, 3, . . .) of all fuzzy hyperedges of the fuzzy hypergraph. ≈ Output: Minimum weighted fuzzy hyperpath (W min ) among all fuzzy hyperpaths from source to destination. ≈ Step 1.: Find all fuzzy hyperpaths and compute weights (W i ) (i = 1, 2, 3, . . .) of each fuzzy hyperpaths by the rule of addition of two IVTFNs. 58 TARASANKAR PRAMANIK AND MADHUMANGAL PAL ≈ Let each W i is of the form ≈ U U U U L L L W i = [(a1 L i , a2 i , a3 i , a4 i ; w W f U )], i = 1, 2, 3, . . . . f L ), (a1 i , a2 i , a3 i , a4 i ; wW i i Step 2.: Initialize ≈ W min U U U U [(a1 L , a2 L , a3 L , a4 L ; wW f L ), (a1 , a2 , a3 , a4 ; wW f U )] = ≈ L L L U U U U W 1 = [(a1 L 1 , a2 1 , a3 1 , a4 1 ; w W gL ), (a1 1 , a2 1 , a3 1 , a4 1 ; wW gU )]. = 1 1 Step 3.: Set i = 2. U U U U Step 4.: Compute [(a1 L , a2 L , a3 L , a4 L ; wW f U )] of each f L ), (a1 , a2 , a3 , a4 ; wW fuzzy hyperpath as a1 L = min(a1 L , a1 L i ) ( L a2 , L L a2 L = a2 L a2 L i −a1 a1 i L a3 = ( L L (a2 L +a2 L i )−(a1 +a1 i ) if a2 L ≤ a1 L i , a3 L , L L a3 L a3 L i −a1 a1 i L +a L ) , (a3 L +a3 L )−(a 1 1i i if a2 L > a1 L i , if a3 L ≤ a1 L i if a3 L > a1 L i , a4 L = min(a4 L , a3 L i ) wW f L = min{wW f L , wW f L} i and U U a1 U 1 = min(a1 , a1 i ) ( U a2 , L U a2 U = a2 U a2 U i −a1 a1 i a3 U = ( U U (a2 U +a2 U i )−(a1 +a1 i ) if a2 U ≤ a1 U i , a3 U , if a3 U ≤ a1 U i U U U a3 a 3 U i −a1 a1 i U U (a3 U +a3 U i )−(a1 +a1 i ) if a2 U > a1 U i , if a3 U > a1 U i a4 U = min(a4 U , a3 U i ) wW f U = min{wW f U , wW fU} i ≈ U U U U Step 5.: Set W min = [(a1 L , a2 L , a3 L , a4 L ; wW f U )] as f L ), (a1 , a2 , a3 , a4 ; wW calculated in Step 4. Step 6.: Increase the value of i by 1. Step 7.: If i < n + 1 go to Step 4, otherwise stop the procedure. end MLIVFH. FUZZY SHORTEST PATH IN IVFHG 59 Illustrative example for the Algorithm MLIVFH:. Consider the hypernetwork shown in Figure 5. Now the weights of all hyperarcs of this hypernetwork are taken as follows: e1 : [(0.4, 0.6, 0.7, 0.8; 0.7), (0.3, 0.5, 0.7, 0.9; 0.8)], e2 : [(0.4, 0.8, 0.10, 0.12; 0.6), (0.2, 0.8, 0.13, 0.15; 0.8)], e3 : [(0.9, 0.15, 0.16, 0.22; 0.6), (0.7, 0.13, 0.17, 0.23; 0.7)], e4 : [(0.13, 0.16, 0.18, 0.27; 0.8), (0.10, 0.12, 0.19, 0.28; 0.87)], e5 : [(0.11, 0.11, 0.21, 25; 0.82), (0.1, 0.1, 21, 0.26; 0.9)], e6 : [(0.9, 0.17, 0.19, 0.23; 0.87), (0.6, 0.12, 0.21, 0.25; 0.92)], e7 : [(0.6, 0.8, 0.11, 0.15; 0.5), (0.3, 0.5, 0.12, 0.17; 0.7)], e8 : [(0.3, 0.5, 0.7, 0.11; 0.65), (0.1, 0.4, 0.11, 0.18; 0.78)]. In Section 3 we see that all the hyperpaths are P-1: 1 → 2 → 6 → 8, P-2: 1 → 3 → 6 → 8, P-3: 1 → 3 → 7 → 8, P-4: 1 → 4 → 7 → 8, P-5: 1 → 5 → 7 → 8. The weights of these hyperpaths are respectively ≈ P-1: W 1 = [(0.21, 0.25, 0.37, 0.48; 0.87), (0.13, 0.18, 0.40, 0.52; 0.92)], ≈ P-2: W 2 = [(0.21, 0.27, 0.40, 0.52; 0.87), (0.12, 0.21, 0.42, 0.57; 0.92)], ≈ P-3: W 3 = [(0.16, 0.30, 0.36, 0.45; 0.7), (0.9, 0.22, 0.40, 0.58; 0.8)], ≈ P-4: W 4 = [(0.18, 0.28, 0.33, 0.45; 0.8), (0.11, 0.22, 0.39, 0.56; 0.87)], ≈ P-5: W 5 = [(0.22, 0.29, 0.35, 0.52; 0.82), (0.14, 0.21, 0.41, 0.63; 0.9)]. ≈ ≈ Now, the value of W min is given by W min = [(0.16, 0.21, 0.22, 0.33; 0.7), (0.9, 0.13, 0.17, 0.39; 0.8)]. 4. C OMPUTATION OF FUZZY SHORTEST HYPERPATH ≈ ≈ In this section, we compute similarity measures [12] S(A, B) to compare two IVTFNs ≈ ≈ A and B. An algorithm is designed to find the fuzzy shortest hyperpath using similarity measurements. Algorithm FSHP Input:: An interval-valued fuzzy hypergraph with a source node and a destination node. Output:: Fuzzy shortest hyperpath of the given interval-valued fuzzy hypergraph. Step 1.: Find out all possible fuzzy hyperpaths from source node to destination node by the Algorithm PF. ≈ Step 2.: Compute W min by using Algorithm MLIVFH. ≈ ≈ Step 3.: Find the similarity measures S(W i , W min ) for i = 1, 2, . . . , p (p being the ≈ ≈ number of all possible fuzzy hyperpaths) of W i and W min . Step 5.: Decide k-th hyperpath as the fuzzy shortest hyperpath having the highest ≈ ≈ similarity measure S(W k , W min ) among all i’s. 60 TARASANKAR PRAMANIK AND MADHUMANGAL PAL end FSHP. Correctness and time complexity of the Algorithm FSHP:. Step 1 of Algorithm FSHP ≈ determines all the hyperpaths of a given hypergraph. Now, W min is minimum among all IVTFNs associated to hyperpaths. By calculating similarity measures between two IVTFNs we have decided that which IVTFN is nearly equal to the minimum most IVTFN ≈ W min and hence it is claimed that the hyperpath associating that IVTFN is the best shortest path. Theorem 4.1. The Algorithm FSHP runs in O(mn) time, where m is the number of edges and n is the number of vertices. Proof. Step 1 takes O(mn) time. Algorithm MLIVFH takes O(m) time as number of paths cannot be greater than the number of edges in a hypergraph. By the same reason Step 3, Step 4 and Step 5 takes O(m) time. So, overall worse case time complexity of the Algorithm FSHP is O(mn). 5. A PPLICATION TO FIND FUZZY SHORTEST HYPERPATH IN RAILWAYS NETWORK Here, we have considered the railways network to find the shortest time require to traverse from a source station to a destination station. The railways networks are connected through more than 7000 stations in India although, we consider a simple structure to understand the work presented in this paper. Assume there are 11 stations A, B, C, D, E, F , G, H, I, J, K. In hypergraph model we take these stations as vertices of the hypergraph and each train as a hyperedge. Since, a train can traverse more than two stations, so this is a hypergraph. The hypergraph of this proposed model is shown in Figure 6. In Figure 6, it is seen that, there are five trains and these trains traverse the stations A, B, C, D, E; B, C, F , G; D, E, H, I, J; G, H, K; J, K. Depending on the real phenomenon, the time required to traverse a train can be considered as fuzzy number. To generalize the proposed problem, here we have considered the time required to traverse the stations is interval-valued trapezoidal fuzzy number. Now, to find the shortest time require to traverse the destination from a source station, hypergraph model is drawn in normal graph model as shown in Figure 7. F IGURE 6. Hypergraph of railways network The time (in hrs.) required to traverse the train between the stations are given in Table 1. For computations, time is converted in terms of 100. FUZZY SHORTEST PATH IN IVFHG 61 Edge Edge weight (A, B) [(0.15, 0.16, 0.18, 0.22; 0.8), (0.13, 0.14, 0.19, 0.23; 0.9)] (A, C) [(0.18, 0.19, 0.20, 0.21; 0.7), (0.14, 0.17, 0.22, 0.25; 0.9)] (A, D) [(0.16, 0.17, 0.18, 0.19; 0.6), (0.15, 0.16, 0.19, 0.22; 0.9)] (A, E) [(0.15, 0.17, 0.19, 0.20; 0.7), (0.12, 0.13, 0.19, 0.22; 0.8)] (B, C) [(0.17, 0.18, 0.20, 0.22; 0.8), (0.16, 0.17, 0.22, 0.23; 0.9)] (C, D) [(0.16, 0.18, 0.19, 0.20; 0.8), (0.15, 0.16, 0.19, 0.21; 0.9)] (D, E) [(0.15, 0.16, 0.17, 0.20; 0.5), (0.14, 0.16, 0.19, 0.20; 0.7)] (B, D) [(0.33, 0.36, 0.39, 0.42; 0.8), (0.31, 0.33, 0.41, 0.44; 0.9)] (B, E) [(0.48, 0.52, 0.56, 0.62; 0.8), (0.45, 0.49, 0.60, 0.63; 0.9)] (B, F ) [(0.18, 0.19, 0.21, 0.22; 0.6), (0.14, 0.16, 0.20, 0.23; 0.9)] (B, G) [(0.19, 0.20, 0.21, 0.22; 0.7), (0.16, 0.18, 0.22, 0.23; 0.8)] (C, F ) [(0.17, 0.18, 0.19, 0.21; 0.5), (0.12, 0.14, 0.19, 0.23; 0.7)] (C, G) [(0.14, 0.15, 0.18, 0.20; 0.7), (0.13, 0.14, 0.19, 0.22; 0.8)] (D, H) [(0.15, 0.16, 0.19, 0.23; 0.8), (0.12, 0.15, 0.20, 0.25; 0.9)] (D, I) [(0.17, 0.18, 0.19, 0.25; 0.5), (0.14, 0.16, 0.22, 0.27; 0.8)] (D, J) [(0.16, 0.17, 0.18, 0.20; 0.6), (0.13, 0.14, 0.19, 0.21; 0.7)] (E, H) [(0.16, 0.17, 0.18, 0.20; 0.6), (0.13, 0.14, 0.19, 0.21; 0.7)] (E, I) [(0.18, 0.19, 0.20, 0.22; 0.7), (0.16, 0.17, 0.22, 0.23; 0.8)] (E, J) [(0.16, 0.17, 0.19, 0.20; 0.8), (0.13, 0.14, 0.19, 0.21; 0.9)] (F, G) [(0.15, 0.17, 0.18, 0.21; 0.6), (0.13, 0.14, 0.19, 0.21; 0.8)] (H, I) [(0.16, 0.18, 0.18, 0.20; 0.5), (0.13, 0.16, 0.19, 0.21; 0.7)] (I, J) [(0.14, 0.16, 0.18, 0.20; 0.7), (0.13, 0.14, 0.19, 0.21; 0.8)] (H, J) [(0.30, 0.34, 0.36, 0.40; 0.7), (0.26, 0.30, 0.38, 0.42; 0.8)] (G, K) [(0.16, 0.17, 0.18, 0.21; 0.6), (0.13, 0.14, 0.19, 0.21; 0.7)] (H, K) [(0.16, 0.18, 0.19, 0.22; 0.6), (0.13, 0.14, 0.19, 0.23; 0.8)] (J, K) [(0.16, 0.17, 0.18, 0.20; 0.6), (0.13, 0.15, 0.19, 0.21; 0.7)] TABLE 1. Edge weights of the graph shown in Figure 7 F IGURE 7. Graph of railways network to find the shortest time to traverse the station K from the station A Now, among all the hyperpaths the shortest paths and their weights between the stations A and K are as follows: ≈ P-1: A −→ B −→ G −→ K; W 1 = [(0.50, 0.53, 0.57, 0.65; 0.8), (0.42, 0.46, 0.60, 0.67; 0.9)] 62 TARASANKAR PRAMANIK AND MADHUMANGAL PAL ≈ P-2: A −→ C −→ G −→ K; W 2 = [(0.48, 0.51, 0.56, 0.62; 0.7), (0.40, 0.45, 0.60, 0.68; 0.9)] ≈ P-3: A −→ D −→ H −→ K; W 3 = [(0.47, 0.51, 0.56, 0.64; 0.8), (0.40, 0.45, 0.58, 0.70; 0.9)] ≈ P-4: A −→ D −→ J −→ K; W 4 = [(0.48, 0.51, 0.54, 0.59; 0.6), (0.41, 0.45, 0.57, 0.64; 0.9)] ≈ P-5: A −→ E −→ H −→ K; W 5 = [(0.47, 0.52, 0.56, 0.62; 0.7), (0.38, 0.41, 0.57, 0.66; 0.8)] ≈ P-6: A −→ E −→ J −→ K; W 6 = [(0.47, 0.51, 0.56, 0.60; 0.8), (0.38, 0.42, 0.57, 0.64; 0.9)] ≈ After usual computations, one can find W min = [(0.47, 0.48, 0.49, 0.54; 0.6), (0.38, 0.38, 0.42, 0.57; 0.8)]. ≈ ≈ ≈ ≈ Routine computations can be done and the results are S(W 1 , W min ) = 0.87, S(W 2 , W min ) = ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ 0.46, S(W 3 , W min ) = 0.88, S(W 4 , W min ) = 0.90, S(W 5 , W min ) = 0.47, S(W 6 , W min ) = 0.88. So, the shortest path from A to K is A −→ D −→ J −→ K. 6. C ONCLUSION Several methods have been found in literature to find the fuzzy shortest path in a hypernetwork. Here we proposed a method using similarity measure to find the fuzzy shortest path in a network with imprecise arc lengths which are IVTFNs. Here BFS technique is used to find all hyperpaths in a hypernetwork. Our proposed algorithm takes O(mn) time, where m and n represent the number of edges and vertices of a fuzzy hypernetwork. Since the IVTFN is more general fuzzy number, our algorithm can be used to solve more generalized shortest path problem on a fuzzy hypernetwork. 7. ACKNOWLEDGMENT We thank reviewers and editors of the journal entitled “Annals of Communications in Mathematics” for their comments that greatly improved the manuscript. R EFERENCES [1] M. Akram and B. Davvaz. Strong intuitionistic fuzzy graphs, Filomat, 26(1)(2012) 177-196. [2] M. Akram, A. Ashraf, and S. Swrwar. 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TARASANKAR P RAMANIK D EPARTMENT OF M ATHEMATICS , K HANPUR G ANGCHE H IGH S CHOOL , PASCHIM M EDINIPUR , 721201, I NDIA Email address: tarasankar.math07@gmail.com M ADHUMANGAL PAL D EPARTMENT OF A PPLIED M ATHEMATICS WITH O CEANOLOGY AND C OMPUTER P ROGRAMMING , V IDYASAGAR U NIVERSITY, M IDNAPORE , 721102, I NDIA Email address: mmpalvu@gmail.com ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 1, Number 1 (2018), 65-73 ISSN: 2582-0818 c http://www.technoskypub.com CUBIC SUBALGEBRAS OF BCH-ALGEBRAS TAPAN SENAPATI∗ AND K. P. SHUM A BSTRACT. In this paper, the notion of cubic subalgebras of BCH-algebras are introduced. Some characterization of cubic subalgebras of BCH-algebras are given. The homomorphic image and inverse image of cubic subalgebras are studied and investigated some related properties. 1. I NTRODUCTION Extending the concept of fuzzy sets (FSs), many scholars introduced various notions of higher-order FSs. Among them, interval-valued fuzzy sets (IVFSs) provides with a flexible mathematical framework to cope with imperfect and imprecise information. Moreover, Jun et al. [14] introduced the concept of cubic sets, as a generalization of fuzzy set and intervalvalued fuzzy set. Jun et al. [15] applied the notion of cubic sets to a group, and introduced the notion of cubic subgroups. Also, Muhiuddin et al. applied the notion of cubic sets to BCK/BCI-algebras on different aspects (see for e.g., [17], [18], [19]). The notions of BCK/BCI-algebras [11] were initiated by Imai and Iseki in 1966 as a generalization of the concept of set-theoretic difference and propositional calculus. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. Senapati et al. [23, 24, 25, 26, 27, 28] have done lot of works on these algebras. In 1983, Hu and Li [9, 10] introduced the notion of a BCH-algebra, which is a generalization of the notions of BCK and BCI-algebras. They have studied a few properties of these algebras. Certain other properties have been studied by Ahmad [1], Chaudhry [5], Chaudhry et al. [6], Dudek and Thomys [8], Roh et. al. [20, 21] and Dar et al. [7]. Based on (bipolar) fuzzy set Jun et al. [12, 13] introduced fuzzy closed ideals and (bipolar) fuzzy filters in BCH-algebras. Using the algebraic structure of soft sets, Kazanc et al. [16] introduced soft BCH-algebras and some of their properties and structural characteristics are discussed and studied. Borumand Saeid et al. [3, 4] introduced the concept of Smarandache BCHalgebras and fuzzy n-fold ideals in BCH-algebras and investigated some of their useful properties. The objective of this paper is to introduce the concept of cubic set to subalgebras of BCH-algebras. The notion of cubic subalgebras of BCH-algebras are defined and lot of 2010 Mathematics Subject Classification. 06F35, 03G25, 08A72. Key words and phrases. BCH-algebra; Subalgebra; Cubic set; Cubic subalgebra. *Corresponding author. 65 66 TAPAN SENAPATI AND K. P. SHUM properties are investigated. Section 2 recalls some definitions, viz., BG-algebra, subalgebra and refinement of unit interval. In Section 3, subalgebras of cubic sets are defined with some its properties. In Section 4, homomorphism of cubic subalgebras and some of its properties are studied. In Section 5, a conclusion of the proposed work is given. 2. P RELIMINARIES An algebra (X, ∗, 0) of type (2, 0) is called a BCH-algebra [9] if it satisfies the following axioms, for all x, y, z ∈ X 1. x ∗ x = 0 2. x ∗ y = 0 and y ∗ x = 0 imply x = y, 3. (x ∗ y) ∗ z = (x ∗ z) ∗ y. Any BCH-algebra X satisfies the following axioms: (i) x ∗ 0 = x, (ii) (x ∗ (x ∗ y)) ∗ y = 0, (iii) 0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y), (iv) 0 ∗ (0 ∗ (0 ∗ x)) = 0 ∗ x, (v) x ≤ y implies 0 ∗ x = 0 ∗ y, for all x, y, z ∈ X [6]. A non-empty subset S of a BG-algebra X is called a subalgebra of X if x ∗ y ∈ S, for all x, y ∈ S. A mapping f : X → Y of BCH-algebras is called a homomorphism if f (x ∗ y) = f (x) ∗ f (y) for all x, y ∈ X. Note that if f : X → Y is a homomorphism, then f (0) = 0. We now review some fuzzy logic concepts as follows: Let X be the collection of objects denoted generally by x. Then a fuzzy set [29] A in X is defined as A = {< x, µA (x) >: x ∈ X} where µA (x) is called the membership value of x in A and 0 ≤ µA (x) ≤ 1. An interval-valued fuzzy set [30] A over X is an object having the form A = {hx, µ̃A (x)i : x ∈ X}, where µ̃A (x) : X → D[0, 1], where D[0, 1] is the set of all subintervals of [0, 1]. The intervals µ̃A (x) denote the intervals of the degree of membership of the element x to + the set A, where µ̃A (x) = [µ− A (x), µA (x)] for all x ∈ X. The determination of maximum and minimum between two real numbers is very simple but it is not simple for two intervals. Biswas [2] described a method to find max/sup and min/inf between two intervals or a set of intervals. + Definition 2.1. [2] Consider two elements D1 , D2 ∈ D[0, 1]. If D1 = [a− 1 , a1 ] and + + + − − D2 = [a− 2 , a2 ], then rmin(D1 , D2 ) = [min(a1 , a2 ), min(a1 , a2 )] which is denoted by − + r D1 ∧ D2 . Thus, if Di = [ai , ai ] ∈ D[0, 1] for i=1,2,3,4,. . . , then we define rsupi (Di ) = + − + − − r [sup(a− i ), sup(ai )], i.e, ∨i Di = [∨i ai , ∨i ai ]. Now we call D1 ≥ D2 iff a1 ≥ a2 and i i + a+ 1 ≥ a2 . Similarly, the relations D1 ≤ D2 and D1 = D2 are defined. Based on the (interval valued) fuzzy sets, Jun et al. [14] introduced the notion of (internal, external) cubic sets, and investigated several properties. Definition 2.2. [14] Let X be a nonempty set. A cubic set A in X is a structure A = {hx, µ̃A (x), νA (x)i : x ∈ X} which is briefly denoted by A = (µ̃A , νA ) where µ̃A = + [µ− A , µA ] is an IVFS in X and νA is a fuzzy set in X. CUBIC SUBALGEBRAS OF BCH-ALGEBRAS 67 3. C UBIC SUBALGEBRAS OF BCH - ALGEBRAS In what follows, let X denote a BCH-algebra unless otherwise specified. Combined the definitions of subalgebras over crisp set and the idea of cubic set, cubic subalgebras of BCH-algebras are defined below. Definition 3.1. Let A = (µ̃A , νA ) be cubic set in X, where X is a subalgebra, then the set A is cubic subalgebra over the binary operator ∗ if it satisfies the following conditions: (F1) µ̃A (x ∗ y) ≥ rmin{µ̃A (x), µ̃A (y)} (F2) νA (x ∗ y) ≤ max{νA (x), νA (y)} for all x, y ∈ X. Let us illustrate this definition using the following example. Example 3.2. Let X={0, a, b, c, d} be a BCH-algebra with the following Cayley table: ∗ 0 a b c d 0 0 a b c d a 0 0 b c d b d d 0 b c c c c d 0 b Define a cubic set A = (µ̃A , νA ) in X by   [0.5, 0.7], if x = 0 [0.3, 0.6], if x = a, c and µ̃A (x) =  [0.2, 0.4], if x = b, d d b b c d 0   0.2, 0.5, νA (x) =  0.7, if x = 0 if x = a, c if x = b, d All the conditions of Definition 3.1 have been satisfy by the set A. Thus A = (µ̃A , νA ) is a cubic subalgebra of X. Proposition 3.1. If A = (µ̃A , νA ) is a cubic subalgebra in X, then for all x ∈ X, µ̃A (0) ≥ µ̃A (x) and νA (0) ≤ νA (x). Thus, µ̃A (0) and νA (0) are the upper bounds and lower bounds of µ̃A (x) and νA (x) respectively. Proof. For all x ∈ X, we have, µ̃A (0) = µ̃A (x ∗ x) ≥ rmin{µ̃A (x), µ̃A (x)} = µ̃A (x) and νA (0) = νA (x ∗ x) ≤ max{νA (x), νA (x)} = νA (x). Theorem 3.2. Let A = (µ̃A , νA ) be a cubic subalgebra of X. If there exists a sequence {xn } in X such that lim µ̃A (xn ) = [1, 1] and lim νA (xn ) = 0. Then µ̃A (0) = [1, 1] n→∞ n→∞ and νA (0) = 0. Proof. By Proposition 3.1, µ̃A (0) ≥ µ̃A (x) for all x ∈ X, therefore, µ̃A (0) ≥ µ̃A (xn ) for every positive integer n. Consider, [1, 1] ≥ µ̃A (0) ≥ lim µ̃A (xn ) = [1, 1]. Hence, n→∞ µ̃A (0) = [1, 1]. Again, by Proposition 3.1, νA (0) ≤ νA (x) for all x ∈ X, thus νA (0) ≤ νA (xn ) for every positive integer n. Now, 0 ≤ νA (0) ≤ lim νA (xn ) = 0. Hence, νA (0) = 0. n→∞ Proposition 3.3. If a cubic set A = (µ̃A , νA ) in X is a cubic subalgebra, then for all x ∈ X, µ̃A (0 ∗ x) ≥ µ̃A (x) and νA (0 ∗ x) ≤ νA (x). Proof. For all x ∈ X, µ̃A (0 ∗ x) ≥ rmin{µ̃A (0), µ̃A (x)} = rmin{µ̃A (x ∗ x), µ̃A (x)} ≥ rmin {rmin{µ̃A (x), µ̃A (x)}, µ̃A (x)} = µ̃A (x) and νA (0 ∗ x) ≤ max{νA (0), νA (x)} = νA (x). 68 TAPAN SENAPATI AND K. P. SHUM + µ̃A (x) = [µ− A (x), µA (x)] represents the membership value and νA (x) represents the + non-membership value of x in cubic set A. But µ− A (x), µA (x) are the membership value and νA (x) are the non-membership value of x to some fuzzy sets. These values can also form a fuzzy subalgebra, proved in the following theorem. + Theorem 3.4. A cubic set A = (µ̃A , νA ) in X is a cubic subalgebra of X iff µ− A , µA and νA are fuzzy subalgebras of X. + − Proof. Let µ− A , µA and νA be fuzzy subalgebras of X and x, y ∈ X. Then µA (x ∗ y) ≥ + + + − min{µ− A (x), µA (y)}, µA (x ∗ y) ≥ min{µA (x), µA (y)} and νA (x ∗ y) ≤ max{νA (x), + − + − νA (y)}. Now, µ̃A (x ∗ y) = [µA (x ∗ y), µA (x ∗ y)] ≥ [min{µ− A (x), µA (y)}, min{µA (x), + − + − + µA (y)}] = rmin{[µA (x), µA (x)], [µA (y), µA (y)]} = rmin{µ̃A (x), µ̃A (y)}. Therefore, A is a cubic subalgebra of X. Conversely, assume that, A = (µ̃A , νA ) is a cubic subalgebra of X. For any x, y ∈ X, + − + [µ− A (x ∗ y), µA (x ∗ y)] = µ̃A (x ∗ y) ≥ rmin{µ̃A (x), µ̃A (y)} = rmin{[µA (x), µA (x)], − + + − − + − [µA (y), µA (y)] = [min{µA (x), µA (y)}, min{µA (x), µA (y)}]. Thus µA (x ∗ y) ≥ min − + + + {µ− A (x), µA (y)}, µA (x∗y) ≥ min{µA (x), µA (y)} and νA (x∗y) ≤ max{νA (x), νA (y)}. + − Hence, µA , µA and νA are fuzzy subalgebras of X. Theorem 3.5. Let A = (µ̃A , νA ) be a cubic subalgebra of X and let n ∈ N (the set of natural numbers). Then n Q (i) µ̃A ( x ∗ x) ≥ µ̃A (x), for any odd number n, n Q (ii) νA ( x ∗ x) ≤ νA (x), for any odd number n, n Q (iii) µ̃A ( x ∗ x) = µ̃A (x), for any even number n, n Q (iv) νA ( x ∗ x) = νA (x), for any even number n. Proof. Let x ∈ X and assume that n is odd. Then n = 2p − 1 for some positive integer p. We prove the theorem by induction. Now µ̃A (x ∗ x) = µ̃A (0) ≥ µ̃A (x) and νA (x ∗ x) = νA (0) ≤ νA (x). Suppose that 2(p+1)−1 2p−1 2p−1 Q Q Q x∗ µ̃A ( x∗x) ≥ µ̃A (x) and νA ( x∗x) ≤ νA (x). Then by assumption, µ̃A ( 2p−1 2p−1 2p+1 Q Q Q x ∗ x) ≥ µ̃A (x) and x ∗ (x ∗ (x ∗ x))) = µ̃A ( x ∗ x) = µ̃A ( x) = µ̃A ( 2(p+1)−1 2p−1 2p−1 2p+1 Q Q Q Q νA ( x∗x) ≤ νA (x), x∗(x∗(x∗x))) = νA ( x∗x) = νA ( x∗x) = νA ( which proves (i) and (ii). Proofs are similar for the cases (iii) and (iv). The sets {x ∈ X : µ̃A (x) = µ̃A (0)} and {x ∈ X : νA (x) = νA (0)} are denoted by Iµ̃A and IνA respectively. These two sets are also subalgebra of X. Theorem 3.6. Let A = (µ̃A , νA ) be a cubic subalgebra of X, then the sets Iµ̃A and IνA are subalgebras of X. Proof. Let x, y ∈ Iµ̃A . Then µ̃A (x) = µ̃A (0) = µ̃A (y) and so, µ̃A (x∗y) ≥ rmin{µ̃A (x), µ̃A (y)} = µ̃A (0). By using Proposition 3.1, we know that µ̃A (x ∗ y) = µ̃A (0) or equivalently x ∗ y ∈ Iµ̃A . Again, let x, y ∈ IνA . Then νA (x) = νA (0) = νA (y) and so, νA (x ∗ y) ≤ max{νA (x), νA (y)} = νA (0). Again, by Proposition 3.1, we know that νA (x ∗ y) = νA (0) or equivalently x ∗ y ∈ IνA . Hence, the sets Iµ̃A and IνA are subalgebras of X. CUBIC SUBALGEBRAS OF BCH-ALGEBRAS 69 Theorem 3.7. Let Bbe a nonempty subset of X and A = (µ̃A , νA ) be cubic set in X [α1 , α2 ], if x ∈ B γ, if x ∈ B defined by µ̃A (x) = and νA (x) = [β1 , β2 ], otherwise δ, otherwise for all [α1 , α2 ], [β1 , β2 ] ∈ D[0, 1] and γ, δ ∈ [0, 1] with [α1 , α2 ] ≥ [β1 , β2 ] and γ ≤ δ. Then A is a cubic subalgebra of X if and only if B is a subalgebra of X. Moreover, Iµ̃A = B = IνA . Proof. Let A be a cubic subalgebra of X. Let x, y ∈ X be such that x, y ∈ B. Then µ̃A (x ∗ y) ≥ rmin{µ̃A (x), µ̃A (y)} = rmin{[α1 , α2 ], [α1 , α2 ]} = [α1 , α2 ] and νA (x ∗ y) ≤ max{νA (x), νA (y)} = max{γ, γ} = γ. So x ∗ y ∈ B. Hence, B is a subalgebra of X. Conversely, suppose that B is a subalgebra of X. Let x, y ∈ X. Consider two cases Case (i) If x, y ∈ B then x ∗ y ∈ B, thus µ̃A (x ∗ y) = [α1 , α2 ] = rmin{µ̃A (x), µ̃A (y)} and νA (x ∗ y) = γ = max{νA (x), νA (y)}. Case (ii) If x ∈ / B or, y ∈ / B, then µ̃A (x ∗ y) ≥ [β1 , β2 ] = rmin{µ̃A (x), µ̃A (y)} and νA (x ∗ y) ≤ δ = max{νA (x), νA (y)}. Hence, A is a cubic subalgebra of X. Now, Iµ̃A = {x ∈ X, µ̃A (x) = µ̃A (0)} = {x ∈ X, µ̃A (x) = [α1 , α2 ]} = B and IνA = {x ∈ X, νA (x) = νA (0)} = {x ∈ X, νA (x) = γ} = B. Definition 3.3. Let A = (µ̃A , νA ) be a cubic set in X. For [s1 , s2 ] ∈ D[0, 1] and t ∈ [0, 1], the set U (µ̃A : [s1 , s2 ]) = {x ∈ X : µ̃A (x) ≥ [s1 , s2 ]} is called upper [s1 , s2 ]-level of A and L(νA : t) = {x ∈ X : νA (x) ≤ t} is called lower t-level of A. Theorem 3.8. If A = (µ̃A , νA ) is a cubic subalgebra of X, then the upper [s1 , s2 ]-level and lower t-level of A are subalgebras of X. Proof. Let x, y ∈ U (µ̃A : [s1 , s2 ]). Then µ̃A (x) ≥ [s1 , s2 ] and µ̃A (y) ≥ [s1 , s2 ]. It follows that µ̃A (x ∗ y) ≥ rmin{µ̃A (x), µ̃A (y)} ≥ [s1 , s2 ] so that x ∗ y ∈ U (µ̃A : [s1 , s2 ]). Hence, U (µ̃A : [s1 , s2 ]) is a subalgebra of X. Let x, y ∈ L(νA : t). Then νA (x) ≤ t and νA (y) ≤ t. It follows that νA (x ∗ y) ≤ max{νA (x), νA (y)} ≤ t so that x ∗ y ∈ L(νA : t). Hence, L(νA : t) is a subalgebra of X. Theorem 3.9. Let A = (µ̃A , νA ) be a cubic set in X, such that the sets U (µ̃A : [s1 , s2 ]) and L(νA : t) are subalgebras of X for every [s1 , s2 ] ∈ D[0, 1] and t ∈ [0, 1]. Then A = (µ̃A , νA ) is a cubic subalgebra of X. Proof. Let for every [s1 , s2 ] ∈ D[0, 1] and t ∈ [0, 1], U (µ̃A : [s1 , s2 ]) and L(νA : t) are subalgebras of X. In contrary, let x0 , y0 ∈ X be such that µ̃A (x0 ∗ y0 ) < rmin{µ̃A (x0 ), µ̃A (y0 )}. Let µ̃A (x0 ) = [θ1 , θ2 ] , µ̃A (y0 ) = [θ3 , θ4 ] and µ̃A (x0 ∗ y0 ) = [s1 , s2 ]. Then [s1 , s2 ] < rmin{[θ1 , θ2 ], [θ3 , θ4 ]} = [min{θ1 , θ3 }, min{θ 2 , θ4 }]. So, s1 < min{θ1 , θ3 } h 1 and s2 < min{θ2 , θ4 }. Let us consider, [ρ1 , ρ2 ] = 2 µ̃A (x0 ∗ y0 ) + rmin{µ̃A (x0 ), µ̃A h i h i (y0 )} = 12 [s1 , s2 ] + [min{θ1 , θ3 }, min{θ2 , θ4 }] = 12 (s1 + min{θ1 , θ3 }), 21 (s2 + i min{θ2 , θ4 }) . Therefore, min{θ1 , θ3 } > ρ1 = 12 (s1 + min{θ1 , θ3 }) > s1 and min{θ2 , θ4 } > ρ2 = 21 (s2 + min{θ2 , θ4 }) > s2 . Hence, [min{θ1 , θ3 }, min{θ2 , θ4 }] > [ρ1 , ρ2 ] > [s1 , s2 ], so that x0 ∗ y0 ∈ / U (µ̃A : [s1 , s2 ]) which is a contradiction, since µ̃A (x0 ) = [θ1 , θ2 ] ≥ [min{θ1 , θ3 }, min{θ2 , θ4 }] > [ρ1 , ρ2 ] and µ̃A (y0 ) = [θ3 , θ4 ] ≥ [min{θ1 , θ3 }, min{θ2 , θ4 }] > [ρ1 , ρ2 ]. This implies x0 ∗ y0 ∈ U (µ̃A : [s1 , s2 ]). Thus µ̃A (x ∗ y) ≥ rmin{µ̃A (x), µ̃A (y)} for all x, y ∈ X. 70 TAPAN SENAPATI AND K. P. SHUM Again, let x0 , y0 ∈ X be such that νA (x0 ∗y0 ) > max{νA (x0 ), νA (y0 )}. Let νA (x0 ) = η1 , νA (y0 ) = η2 and νA (x0 ∗ y0 ) = t. Then t > max{η1 , η2 }. Let us consider, t1 = 1 1 2 [νA (x0 ∗ y0 ) + max{νA (x0 ), νA (y0 )}]. We get that t1 = 2 (t + max{η1 , η2 }). Therefore, 1 1 η1 < t1 = 2 (t + max{η1 , η2 }) < t and η2 < t1 = 2 (t + max{η1 , η2 }) < t. Hence, max{η1 , η2 } < t1 < t = νA (x0 ∗ y0 ), so that x0 ∗ y0 ∈ / L(νA : t) which is a contradiction, since νA (x0 ) = η1 ≤ max{η1 , η2 } < t1 and νA (y0 ) = η2 ≤ max{η1 , η2 } < t1 . This implies x0 , y0 ∈ L(νA : t). Thus νA (x ∗ y) ≤ max{νA (x), νA (y)} for all x, y ∈ X. Theorem 3.10. Any subalgebra of X can be realized as both the upper [s1 , s2 ]-level and lower t-level of some cubic subalgebra of X. Proof. Let P be a cubic subalgebra of X, and A be cubic set on X defined by β, if x ∈ P [α1 , α2 ], if x ∈ P and νA (x) = µ̃A (x) = 1, otherwise [0, 0], otherwise for all [α1 , α2 ] ∈ D[0, 1] and β ∈ [0, 1]. We consider the following cases: Case (i) If x, y ∈ P , then µ̃A (x) = [α1 , α2 ], νA (x) = β and µ̃A (y) = [α1 , α2 ], νA (y) = β. Thus, µ̃A (x ∗ y) = [α1 , α2 ] = rmin{[α1 , α2 ], [α1 , α2 ]} = rmin{µ̃A (x), µ̃A (y)} and νA (x ∗ y) = β = max{β, β} = max{νA (x), νA (y)}. Case (ii) If x ∈ P and y ∈ / P then µ̃A (x) = [α1 , α2 ], νA (x) = β and µ̃A (y) = [0, 0], νA (y) = 1. Thus, µ̃A (x ∗ y) ≥ [0, 0] = rmin{[α1 , α2 ], [0, 0]} = rmin{µ̃A (x), µ̃A (y)} and νA (x ∗ y) ≤ 1 = max{β, 1} = max{νA (x), νA (y)}. Case (iii) If x ∈ / P and y ∈ P then µ̃A (x) = [0, 0], νA (x) = 1 and µ̃A (y) = [α1 , α2 ], νA (y) = β. Thus, µ̃A (x ∗ y) ≥ [0, 0] = rmin{[0, 0], [α1 , α2 ]} = rmin{µ̃A (x), µ̃A (y)} and νA (x ∗ y) ≤ 1 = max{1, β} = max{νA (x), νA (y)}. Case (iv): If x ∈ / P and y ∈ / P then µ̃A (x) = [0, 0], νA (x) = 1 and µ̃A (y) = [0, 0], νA (y) = 1. Now µ̃A (x ∗ y) ≥ [0, 0] = rmin{[0, 0], [0, 0]} = rmin{µ̃A (x), µ̃A (y)} and νA (x ∗ y) ≤ 1 = max{1, 1} = max{νA (x), νA (y)}. Therefore, A is a cubic subalgebra of X. Theorem 3.11. Let P be a subset of X and A be cubic set on X which is given in the proof of Theorem 3.10. If A be realized as lower level subalgebra and upper level subalgebra of some cubic subalgebra of X, then P is a cubic subalgebra of X. Proof. Let A be a cubic subalgebra of X, and x, y ∈ P . Then µ̃A (x) = [α1 , α2 ] = µ̃A (y) and νA (x) = β = νA (y). Thus µ̃A (x∗y) ≥ rmin{µ̃A (x), µ̃A (y)} = rmin{[α1 , α2 ], [α1 , α2 ]} = [α1 , α2 ] and νA (x ∗ y) ≤ max{νA (x), νA (y)} = max{β, β} = β, which imply that x ∗ y ∈ P . Hence, the theorem. 4. H OMOMORPHISM OF CUBIC SUBALGEBRAS In this section, homomorphism of cubic subalgebra is defined and some results are studied. Let f be a mapping from a set X into a set Y . Let B = (µ̃B , νB ) be cubic set in Y . Then the inverse image of B, is defined as f −1 (B) = {hx, f −1 (µ̃B ), f −1 (νB )i : x ∈ X} with the membership function and non-membership function respectively are given by f −1 (µ̃B )(x) = µ̃B (f (x)) and f −1 (νB )(x) = νB (f (x)). It can be shown that f −1 (B) is cubic set. Theorem 4.1. Let f : X → Y be a homomorphism of BCH-algebras. If B = (µ̃B , νB ) is a cubic subalgebra of Y , then the preimage f −1 (B) = {hx, f −1 (µ̃B ), f −1 (νB )i : x ∈ X} of B under f is a cubic subalgebra of X. CUBIC SUBALGEBRAS OF BCH-ALGEBRAS 71 Proof. Assume that B = (µ̃B , νB ) is a cubic subalgebra of Y and let x, y ∈ X. Then f −1 (µ̃B )(x ∗ y) = µ̃B (f (x ∗ y)) = µ̃B (f (x) ∗ f (y)) ≥ rmin{µ̃B (f (x), µ̃B (f (y))} = rmin{f −1 (µ̃B ) (x), f −1 (µ̃B )(y)} and f −1 (νB )(x ∗ y) = νB (f (x ∗ y)) = νB (f (x) ∗ f (y)) ≤ max{νB (f (x), νB (f (y))} = max{f −1 (νB )(x), f −1 (νB )(y)}. Therefore, f −1 (B) = {hx, f −1 (µ̃B ), f −1 (νB )i : x ∈ X} is a cubic subalgebra of X. Definition 4.1. A cubic set A in the BCH-algebra X is said to have the rsup-property and inf-property if for any subset T of X there exist t0 ∈ T such that µ̃A (t0 ) = rsupt0 ∈T µ̃A (t) and νA (t0 ) = inf νA (t) respectively. t0 ∈T Definition 4.2. Let f be a mapping from the set X to the set Y . If A = (µ̃A , νA ) is cubic set in X, then the image of A under f , denoted by f (A), and is defined as f (A) = {hx, frsup (µ̃A ), finf (ν A )i : x ∈ Y }, where rsupx∈f −1 (y) µ̃A (x), if f −1 (y) 6= φ frsup (µ̃A )(y) = [0, 0], otherwise and ( inf νA (x), if f −1 (y) 6= φ x∈f −1 (y) finf (νA )(y) = 1, otherwise. Theorem 4.2. Let f : X → Y be a homomorphism from a BCH-algebra X onto a BCH-algebra Y . If A = (µ̃A , νA ) is a cubic subalgebra of X, then the image f (A) = {hx, frsup (µ̃A ), finf (νA )i : x ∈ Y } of A under f is a cubic subalgebra of Y . Proof. Let A = (µ̃A , νA ) be a cubic subalgebra of X and let y1 , y2 ∈ Y . We know that, {x1 ∗ x2 : x1 ∈ f −1 (y1 ) and x2 ∈ f −1 (y2 )} ⊆ {x ∈ X : x ∈ f −1 (y1 ∗ y2 )}. Now, frsup (µ̃A )(y1 ∗ y2 ) = rsup{µ̃A (x) : x ∈ f −1 (y1 ∗ y2 )} ≥ rsup{µ̃A (x1 ∗ x2 ) : x1 ∈ f −1 (y1 ) and x2 ∈ f −1 (y2 )} ≥ rsup{rmin{µ̃A (x1 ), µ̃A (x2 )} : x1 ∈ f −1 (y1 ), x2 ∈ f −1 (y2 )} = rmin{rsup{µ̃A (x1 ) : x1 ∈ f −1 (y1 )}, = rsup{µ̃A (x2 ) : x2 ∈ f −1 (y2 )}} = rmin{frsup (µ̃A )(y1 ), frsup (µ̃A )(y2 )} = inf{νA (x) : x ∈ f −1 (y1 ∗ y2 )} ≤ inf{νA (x1 ∗ x2 ) : x1 ∈ f −1 (y1 ) and x2 ∈ f −1 (y2 )} ≤ inf{max{νA (x1 ), νA (x2 )} : x1 ∈ f −1 (y1 ), x2 ∈ f −1 (y2 )} = max{inf{νA (x1 ) : x1 ∈ f −1 (y1 )}, = = inf{νA (x2 ) : x2 ∈ f −1 (y2 )}} max{finf (νA )(y1 ), finf (νA )(y2 )}. and finf (νA )(y1 ∗ y2 ) Hence, f (A) = {hx, frsup (µ̃A ), finf (νA )i : x ∈ Y } is a cubic subalgebra of Y . 72 TAPAN SENAPATI AND K. P. SHUM 5. C ONCLUSIONS To investigate the structure of an algebraic system, it is clear that subalgebras with special properties play an important role. 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Fuzzy dot subalgebras and fuzzy dot ideals of B-algebras, Journal of Uncertain Systems, 8(1) (2014), 22-30. [28] T. Senapati, Y.B. Jun, G. Muhiuddin and K. P. Shum. Cubic intuitionistic structures applied to ideals of BCI-algebras, Analele Univ. Ovidius, Constanta, Ser. Mat., (2019) (In press). [29] L. A. Zadeh. Fuzzy sets, Information and Control, 8 (1965), 338-353. [30] L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning. I, Inform. Sci., 8 (1975), 199-249. TAPAN S ENAPATI D EPARTMENT OF A PPLIED M ATHEMATICS WITH O CEANOLOGY AND C OMPUTER P ROGRAMMING , V IDYASAGAR U NIVERSITY, M IDNAPORE 721102, I NDIA Email address: math.tapan@gmail.com K. P. S HUM I NSTITUTE OF M ATHEMATICS , Y UNNAN U NIVERSITY, K UNMING 650091, P EOPLE ’ S R EPUBLIC OF C HINA Email address: kpshum@ynu.edu.cn ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 1, Number 1 (2018), 74-84 ISSN: 2582-0818 c http://www.technoskypub.com GENERALIZED SYMMETRIC BI-DERIVATIONS OF LATTICES CHIRANJIBE JANA∗ AND MADHUMANGAL PAL A BSTRACT. In this article, the notion of a new kind of derivation is introduced for a lattice L called symmetric bi-(T, F )-derivations on L as a generalization of derivation of lattices and characterized some of its related properties. Some equivalent conditions provided for a lattice L with greatest element 1 by the notion of isotone symmetric bi(T, F )-derivation on L. By using the concept of isotone derivation, we characterized the modular and distributive lattices by the notion of isotone symmetric bi-(T, F )-derivation. 1. I NTRODUCTION The notion of lattice theory first introduced by Birkhof [7]. After the initiation of lattices many researchers studied lattice theory in different point of view such as, Balbes and Dwinger [3] gave the concept on distributive lattices and Hoffmann gave the notion of partially ordered set (Poset). The application of lattice theory plays an important role in different areas such as information theory [4], information retrieval [10], information access controls [39] and cryptanalysis [14]. Recently, the properties of lattices were studied by some authors [13, 16, 26] in analytic and algebraic point of view. Derivations is a very interesting research area in the theory of algebraic structure in mathematics. Posner [37] provided the concept of derivation on rings. Based on this concept Bell and Kappe [5] studied that rings in which derivations satisfy certain algebraic conditions and Kaya [28] applied the notions of derivations on prime rings. The notion of generalized derivation in ring introduced by Braser [8, 9] and Hvala [17]. This concept of derivation further carried out by many authors [2, 15] in prime rings and lie ideal in prime rings. Jun and Xin [25] applied the notion of derivation in ring and near ring theory to BCI-algebra. Later on, Muhiuddin et al. studied the theory of derivations in BCI-algebras on different aspects (see for e.g., [31],[32], [33], [34]). Jana et al. [21-30] and, Bej and Pal [6] and Senapati et al. [40] has done lot of works on BCK/BCI-algebra and B/BG/G-algebras which is related to these algebras. Zhan and Liu [45] studied the notion of left-right (respectively, right-left) f -derivation of BCI-algebras and investigated its properties. The study of derivation in lattice theory is an important topic in application of different mode. Xin et al. [43] introduced the notion of derivation in lattices and discussed its properties. Thereafter, many authors generalized this idea in lattices. For example Yilmaz and Öztürk [44] introduced the notion of f -derivation on lattices and its 2010 Mathematics Subject Classification. 03G16, 06C05, 17A36. Key words and phrases. Lattice; Derivation of lattice; Symmetric bi-(T, F )-derivation of lattice. 74 GENERALIZED SYMMETRIC BI-DERIVATIONS OF LATTICES 75 some related properties discussed, Çeven [12] studied symmetric bi-derivation on lattice, Kim [27] further investigated symmetric bi-f -derivations on lattices, Alshehri [1] studied generalized derivation on lattices and Chaudhry and Ullah [11] introduced the notion of (α, β)-generalized derivations on lattices and some of its related properties investigated. After symmetric bi-derivation studied by Maksa [29, 30], many researchers introduced this concept to study symmetric bi-derivation on rings and near-rings [35, 36, 38, 41, 42]. Recently, Çven [12] studied symmetric bi-derivation on lattices and investigated some properties on it. Motivated by the above works and best of our knowledge there is no work available on symmetric bi-(T, F )-derivations on lattices. For this reason we developed theoretical study of symmetric bi-(T, F )-derivation on lattices. In this paper, the notion of symmetric bi-(T, F )-derivation which is a generalization of derivation in lattices is introduced and studied some properties of it. We gave some equivalent condition for which a derivation to be an isotone symmetric bi-(T, F )-derivation for a lattices with greatest element. We characterized modular lattices and distributive lattices by the concept of isotone symmetric bi-(T, F )-derivation. 2. P RELIMINARIES Definition 2.1. [7] Let L be a non-empty set endowed with operations ∧ and ∨. Then the set (L, ∧, ∨) is called lattices if for all x, y, z ∈ L satisfies the following conditions: (L1) x ∧ x = x, x ∨ x = x (L2) x ∧ y = y ∧ x, x ∨ y = y ∨ x (L3) (x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (L4) (x ∧ y) ∨ x = x, (x ∨ y) ∧ x = x. Definition 2.2. [7] A Lattice (L, ∧, ∨) is called distributive lattice if for all x, y, z ∈ L satisfies the following conditions: (L5) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (L6) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). It is notified that in a Lattice the conditions (L5) and (L6) are equivalent. Definition 2.3. [7] Let (L, ∧, ∨) be a lattice. A binary relations (≤) on L defined by x ≤ y is holds if and only if x ∧ y = x and x ∨ y = y. Definition 2.4. [3] A lattice (L, ∧, ∨) is called a modular lattice if for all x, y, z ∈ L satisfies the following conditions: (L7) If x ≤ y implies x ∨ (y ∧ z) = (x ∨ y) ∧ z. Definition 2.5. [43] Let (L, ∧, ∨) be a lattice. Then (L, ≤) is a poset, i.e. it is a partially ordered set and for any x, y ∈ L, x ∧ y is the g.l.b of {x, y}, and x ∨ y is the l.u.b of {x, y}. Definition 2.6. [7] Let F : L → M be a function from the lattice L to the lattice M and is called lattice homomorphism if it satisfies the conditions: (L10 ) F (x ∧ y) = F (x) ∧ F (y), F (x ∨ y) = F (x) ∨ F (y) for all x, y ∈ L. Definition 2.7. [43] Let L be a lattice and d be a self-map on L. Then for all x, y ∈ L, d is called derivation on L if satisfying the following identity : d(x ∧ y) = (d(x ∧ y) ∨ (x ∧ d(y)) Proposition 2.1. [43] Let L be a lattice and d be a derivation on L. Then following conditions are hold: (1) d(x) ≤ x 76 CHIRANJIBE JANA AND MADHUMANGAL PAL (2) d(x) ∧ d(y) ≤ d(x ∧ y) ≤ d(x) ∨ d(y) (3) If L has a least element 0 and a greatest element 1, then d(0) = 0 and d(1) ≤ 1. Definition 2.8. [43] Let L be a lattice and d be a derivation on L (5) If x ≤ y implies d(x) ≤ d(y), then d is called an isotone derivation (6) If d is one-to-one, then d is called a monomorphic derivation (7) If d is onto, then d is called an epimorphic derivation. Definition 2.9. [12, 41] Let (L, ∧, ∨) be a lattice. A function D : L × L → L is called symmetric if it satisfies the condition D(x, y) = D(y, x) for all x, y ∈ L. Definition 2.10. [35] Let L be a lattice. A function d : L × L → L defined by d(x) = D(x, x) is called trace of D, where D is a symmetric function. Definition 2.11. [12] Let L be a lattice and Let D : L × L → L be a symmetric function on L. Then D is called symmetric bi-derivation on L if it satisfies the following identity: D(x ∧ y, z) = (D(x, z) ∧ y) ∨ (x ∧ D(y, z)) for all x, y, z ∈ L. Also, a symmetric bi-derivation D satisfies the following relation D(x, y ∧ z) = (D(x, y) ∧ z) ∨ (y ∧ D(x, z)) for all x, y, z ∈ L. 3. S YMMETRIC BI -(T, F )- DERIVATIONS ON LATTICES In this section, symmetric bi-(T, F )-derivation on a lattices is introduced. Definition 3.1. Let L be a lattice. Then for any T ∈ L, we define a self-map DT : L×L → L by DT (x, y) = (x ∧ y) ∧ T for all x, y ∈ L. Definition 3.2. Let L be a lattice. Then for any T ∈ L, a self-map DT : L × L → L is defined as for any T ∈ L, DT (x, y) = (x ∧ y) ∧ T for all x ∈ L. Then then function DT : L × L → L is called symmetric bi-(T, F )-derivation of L if there exist a function F : L → L satisfies the condition: DT (x ∧ y, z) = (DT (x, z) ∧ F (y)) ∨ (F (x) ∧ DT (y, z)) for all x, y, z ∈ L. Also, a symmetric bi-(T, F )-derivation DT satisfies the following relation DT (x, y ∧ z) = (DT (x, y) ∧ F (z)) ∨ (F (y) ∧ DT (x, z)) for all x, y, z ∈ L. It is notified in the Definition 3.2 that if F is an identity function then DT is a symmetric bi-T -derivation on L. Therefore, according to Definition 3.2, DT is a symmetric bi-(T, F )-derivation on L if F must satisfied the identity of the Definition 3.2. Example 3.1. Let L = {0, a, b, 1} be a lattice shown by the Hasse diagram of Figure 1 For any T ∈ L, define a self-map DT : L × L → L of a lattice L given in figure 2 Define the mapping DT as follows: for T = 0, DT (x, y) = 0 for all (x, y) ∈ L × L for T = a, DT (x, y) = 0 for all (x, y) ∈ {(0, 0), (0, a), (a, 0), (b, 0), (0, b), (1, 0), (0, 1)} GENERALIZED SYMMETRIC BI-DERIVATIONS OF LATTICES 77 DT (x, y) = a for all (x, y) ∈ {(a, a), (a, b), (b, a), (a, 1), (1, a), (b, b), (b, 1), (1, b), (1, 1)} for T = b, DT (x, y) = 0 for all (x, y) ∈ {(0, 0), (a, 0), (0, a), (0, b), (b, 0), (1, 0), (0, 1)}, DT (x, y) = a for all (x, y) ∈ {(a, a), (a, b), (b, a), (a, 1), (1, a)} and DT (x, y) = b for all (x, y) ∈ {(b, b), (b, 1), (1, b), (1, 1)} For T = 1, DT (x, y) = 0 for all (x, y) ∈ {(0, 0), (0, a), (a, 0), (b, 0), (0, b), (1, 0), (0, 1)}, DT (x, y) = a for all (x, y) ∈ {(a, a), (a, b), (b, a), (a, 1), (1, a), DT (x, y) = b for all (x, y) ∈ {(b, b), (b, 1), (1, b)} and DT (x, y) = 1 for (x, y) = (1, 1). If we defined the function F by F (0) = 0, F (a) = a, F (b) = 1 and F (1) = b, then it is verified that for each T ∈ L, DT is a symmetric bi-(T, F )-derivation on L. Where as, if we defined the function F by F (0) = 0, F (a) = b, F (b) = a and F (1) = 1, then it is justified that for each T ∈ L, DT is not a symmetric bi-(T, F )derivation of L, because for T = b, we have DT (a ∧ b, 1) = DT (a, 1) = (a ∧ 1) ∧ b = a ∧ b = a, but (DT (a, 1) ∧ F (b)) ∨ (F (a) ∧ DT (b, 1)) = ((a ∧ 1) ∧ a) ∨ (b ∧ (b ∧ 1)) = (a ∧ a) ∨ (b ∧ b) = a ∨ b = b. Therefore, DT (a ∧ b, 1) = a 6= b = (DT (a, 1) ∧ F (b)) ∨ (F (a) ∧ DT (b, 1)). ④1 ④ b ⑤a ③ 0 F IGURE 1. The lattice in example 3.3 Example 3.2. Let L be a lattice and T ∈ L. Defining a function DT : L → L by DT (x, y) = (F (x) ∧ F (y)) ∧ T for all x, y ∈ L where F : L → L satisfying F (x ∧ y) = F (x) ∧ F (y) for all x, y ∈ L. Then DT is a symmetric bi-(T, F )-derivation of L. In addition, if F is an increasing function then DT is an isotone symmetric bi-(T, F ) derivation on L. Theorem 3.3. Let L be a lattice and DT be a trace of symmetric bi-(T, F )-derivation DT . Then following conditions are hold for all x, y ∈ L. (1) DT (x, y) ≤ F (x) and DT (x, y) ≤ F (y) (2) DT (x, y) ∧ DT (w, y) ≤ DT (x ∧ w, y) ≤ DT (x, y) ∨ DT (w, y) (3) DT (x ∧ w, y) ≤ F (x) ∨ F (w) (4) DT (x, y) ≤ F (x) ∧ F (y) (5) DT (x) ≤ F (x) (6) d2T (x) = DT (x). 78 CHIRANJIBE JANA AND MADHUMANGAL PAL Proof. (1) Since DT (x, y) = DT (x ∧ x, y) = (DT (x, y) ∧ F (x)) ∨ (F (x) ∧ DT (x, y)) = F (x) ∧ DT (x, y) from which we get DT (x, y) ≤ F (x). Similarly, DT (x, y) ≤ F (y) for all x, y ∈ L. (2) Since DT (x, y) ≤ F (x) and DT (w, y) ≤ F (w). Then, we have DT (x, y)∧DT (w, y) ≤ F (x) ∧ DT (w, y), and from (1) DT (x, y) ∧ DT (w, y) ≤ F (w) ∧ DT (x, y) for all x, y, w ∈ L. Hence, DT (x, y) ∧ DT (w, y) ≤ (F (x) ∧ DT (w, y)) ∨ (F (w) ∧ DT (x, y)) = DT (x ∧ w, y). Also, since F (x) ∧ DT (w, y) ≤ DT (w, y) and F (w) ∧ DT (x, y) ≤ DT (x, y), and hence obtained (F (x) ∧ DT (w, y)) ∨ (F (w) ∧ DT (x, y) ≤ DT (x, y) ∨ DT (w, y). Thus, DT (x ∧ w, y) ≤ DT (x, y) ∨ DT (w, y). (3) Since DT (x, y)∧F (w) ≤ F (w) and F (x)∧DT (w, y) ≤ F (x). Therefore, (DT (x, y)∧ F (w)) ∨ (F (x) ∧ DT (w, y)) ≤ F (x) ∨ F (w). Hence, DT (x ∧ w, y) ≤ F (x) ∨ F (w). (4) From (1) it is clear that DT (x, y) ≤ F (x) ∧ F (y) for all x, y ∈ L. (5) Since DT (x) = DT (x ∧ x, x) = (DT (x, x) ∧ F (x)) ∨ (F (x) ∧ DT (x, x)) = F (x) ∧ DT (x, y) from which we obtained DT (x) ≤ F (x) for all x ∈ L. (6) From (5) it is seen that d2T (x) = DT (DT (x)) ≤ DT (x) ≤ F (x) and also from (A) gives DT (x, DT (x)) ≤ DT (x). Then, we have d2T (x) = DT (DT (x)) = DT (F (x) ∧ DT (x)) = DT (F (x), DT (x)) ∨ (F (x) ∧ d2T (x)) ∨ (DT (x) ∧ F (x)) = DT (F (x), DT (x)) ∨ d2T (x) ∨ DT (x) = DT (F (x), DT (x)) ∨ DT (x). Corollary 3.1. Let L be a lattice and DT be a symmetric bi-(T, F )-derivation on L with least element 0 and greatest element 1, then F (0) = 0 and F (1) = 1 implies DT (0, x) = 0 and DT (1, x) ≤ F (x) for all x ∈ L. Proof: The proof of the corollary is trivial by Theorem 3.3(1). ✷ Theorem 3.4. Let L be a lattice and DT be symmetric bi-(T, F )-derivation of L and DT be the trace of symmetric bi-(T, F )-derivation DT . Then, DT (x ∧ y) = DT (x, y) ∨ (F (x) ∧ DT (y)) ∨ (F (y) ∧ DT (x)) for all x, y ∈ L. Proof. By using the Theorem 3.3 (1) and (5), we have DT (x ∧ y) = DT (x ∧ y, x ∧ y) = = (DT (x ∧ y, x) ∧ F (y)) ∨ (DT (x ∧ y, y) ∧ F (x)) DT (x ∧ y, x) ∨ DT (x ∧ y, y) = ((DT (x) ∧ F (y)) ∨ (F (x) ∧ DT (x, y))) ∨ ((DT (x, y) ∧ F (y)) ∨ (F (x) ∧ DT (y))) = = ((DT (x) ∧ F (y)) ∨ DT (x, y)) ∨ (DT (x, y) ∨ (F (x) ∧ DT (y))) DT (x, y) ∨ (F (x) ∧ DT (y)) ∨ (F (y) ∧ DT (x)). Corollary 3.2. Let L be a lattice and DT be symmetric bi-(T, F )-derivation of L and DT be the trace of symmetric bi-(T, F )-derivation DT . Then the followings inequalities hold: for all x, y ∈ L (1) DT (x, y) ≤ DT (x ∧ y) GENERALIZED SYMMETRIC BI-DERIVATIONS OF LATTICES 79 (2) F (x) ∧ DT (y) ≤ DT (x ∧ y) (3) F (y) ∧ DT (x) ≤ DT (x ∧ y) (4) DT (x) ∧ DT (y) ≤ DT (x ∧ y). Proof. The proof of (1), (2) and (3) are trivial by Theorem 3.4. (4) can be proved by using (2), (3) and Theorem 3.3(5). Proposition 3.5. Let L be a lattice with least element 0 and greatest element 1, and DT be symmetric bi-(T, F )-derivation of L and dT be the trace of symmetric bi-(T, F )-derivation DT , then following results hold: (1) If F (x) ≥ DT (1, y), then DT (x, y) ≥ DT (1, y) (2) If F (x) ≤ DT (1, y), then DT (x) = F (x) Proof. (1) Let F (1) = 1, then DT (x, y) = = DT (x ∧ 1, y) (DT (x, y) ∧ F (1)) ∨ (F (x) ∧ DT (1, y)) = DT (x, y) ∨ DT (1, y). Hence, DT (x, y) ≥ DT (1, y) for all x, y ∈ L (2) DT (x, y) = DT (x ∧ 1, y) = = (DT (x, y) ∧ F (1)) ∨ (F (x) ∧ DT (1, y)) DT (x, y) ∨ F (x). Then, F (x) ≤ DT (x, y). Hence by Theorem 3.3(1), DT (x, y) = F (x) for all x, y ∈ L. Theorem 3.6. Let L be a lattice with greatest element 1 and let DT be a trace of a symmetric bi-(T, F )-derivation DT . Then following conditions are equivalent: (1) DT is an isotone mapping (2) DT (x) = F (x) ∧ DT (1) (3) DT (x ∧ y) = DT (x) ∧ DT (y) (4) DT (x) ∨ DT (y) ≤ DT (x ∨ y). Proof. (1) ⇒ (2). Since DT is isotone and x ≤ 1, then F (x) ≤ DT (1). Also, DT (x) ≤ F (x) ∧ DT (1) by Theorem 3.3(E). By Corollary 3.2 (B), we have F (x) ∧ DT (1) ≤ DT (x) for all x ∈ L. Hence, DT (x) = F (x) ∧ DT (1) for all x ∈ L. (2) ⇒ (3). Let F (x) ∧ DT (1) = DT (x) for all x ∈ L. Then, DT (x ∧ y) = F (x ∧ y) ∧ DT (1) = (F (x) ∧ DT (1)) ∧ (F (y) ∧ DT (1)) = DT (x) ∧ DT (y) for all x, y ∈ L. (3) ⇒ (1). Let DT (x ∧ y) = DT (x) ∧ DT (y) for all x, y ∈ L and x ≤ y. Then, DT (x) = DT (x ∧ y) = DT (x) ∧ DT (y), and hence DT (x) ≤ DT (y). (1) ⇒ (4). Let DT be isotone. Since x ≤ x∨y and y ≤ x∨y, then DT (x) ≤ DT (x∨y) and DT (y) ≤ DT (x ∨ y). Thus, DT (x) ∨ DT (y) ≤ DT (x ∨ y) for all x, y ∈ L. (4) ⇒ (1). Let x ≤ y, then DT (x) ≤ DT (x ∨ y) = DT (y). Hence, DT is isotone. Definition 3.3. Let L be a lattice and DT be a symmetric bi-(T, F )-derivation of L (1) If x ≤ w implies DT (x, y) ≤ DT (w, y), then DT is called an isotone symmetric bi(T, F )-derivation (2) If DT is one-to-one, then DT is called a monomorphic symmetric bi-(T, F )-derivation (3) If DT is onto, then DT is called an epic symmetric bi-(T, F )-derivation. 80 CHIRANJIBE JANA AND MADHUMANGAL PAL Lemma 3.7. Let DT be a symmetric bi-(T, F )-derivation on lattice L. Then followings hold: (1) DT (x ∧ w, y) = DT (x, y) ∧ DT (w, y) for all x, y, w ∈ L (2) DT (x ∨ w, y) ≥ DT (x, y) ∨ DT (w, y) for all x, y, w ∈ L. Proof. (1) Since x ∧ w ≤ x and x ∧ w ≤ w, then we have DT (x ∧ w, y) ≤ DT (x, y) and DT (x ∧ w, y) ≤ DT (w, y). Thus DT (x ∧ w, y) ≤ DT (x, y) ∧ DT (w, y). Hence, by Theorem 3.3(2), we get DT (x ∧ w, y) = DT (x, y) ∧ DT (w, y) for all x, y, w ∈ L. (2) Since x ≤ x ∨ y and y ≤ x ∨ y, so we have DT (x, y) ≤ DT (x ∨ w, y) and DT (w, y) ≤ DT (x ∨ w, y). Therefore, we obtained DT (x ∨ w, y) ≥ DT (x, y) ∨ DT (w, y) for all x, y, w ∈ L. Proposition 3.8. Let L be a lattice and DT be a symmetric bi-(T, F )-derivation on L. Then following conditions hold: (1) DT (x, y) = DT (x, y) ∨ (DT (x ∨ s, y) ∧ x), when DT is an symmetric bi-(T, F )derivation on L (2) DT (x, y) = DT (x, y) ∨ (DT (x ∨ s, y) ∧ F (x)), when F is a join-homomorphism on L (3) Then DT (x, y) = DT (x, y) ∨ (F (x) ∧ DT (x ∨ s, y), when F is an increasing function on L. Proof. (1) Let DT be an isotone symmetric bi-(T, F )-derivation. Then, DT (x, y) = DT ((x ∨ s) ∧ x, y) = = (DT (x ∨ s, y) ∧ F (x)) ∨ (F (x ∨ s) ∧ DT (x, y)) (DT (x ∨ s, y) ∧ F (x)) ∨ DT (x, y). As, DT (x, y) ≤ DT (x ∨ s, y) ≤ F (x ∨ s). (2) Since DT (x, y) ≤ F (x) ≤ F (x) ∨ F (s) and F (x ∨ s) = F (x) ∨ F (s), so obtained DT (x, y) = DT ((x ∨ s) ∧ x, y) = (DT (x ∨ s, y) ∧ F (x)) ∨ (F (x ∨ s) ∧ DT (x, y)) = (DT (x ∨ s, y) ∧ F (x)) ∨ DT (x, y). (3) Since F is an increasing function and x ≤ x ∨ y, so F (x) ≤ F (x ∨ y). Therefore, DT (x, y) = DT ((x ∨ s) ∧ x, y) = = (DT (x ∨ s, y) ∧ F (x)) ∨ (F (x ∨ s) ∧ DT (x, y)) (DT (x ∨ s, y) ∧ F (x)) ∨ DT (x, y). Theorem 3.9. Let L be a lattice with greatest element 1 and DT be a symmetric bi-(T, F )derivation on L and F (x ∧ y) = F (x) ∧ F (y). Then followings equivalent: (1) DT is isotone symmetric bi-(T, F )-derivation (2) DT (x, y) ∨ DT (s, y) ≤ DT (x ∨ s, y) for all x, y ∈ L (3) DT (x, y) = F (x) ∧ DT (1, y) for all x, y ∈ L (4) DT (x ∧ s, y) = DT (x, y) ∧ DT (s, y) for all x, y, s ∈ L. Proof. (1) ⇒ (2). We assume that DT is an isotone symmetric bi-(T, F )-derivation on L. Since x ≤ x ∨ s and s ≤ x ∨ s, and so DT (x, y) ≤ DT (x ∨ s, y) and DT (s, y) ≤ DT (x ∨ s, y). Hence, DT (x, y) ∨ DT (s, y) ≤ DT (x ∨ s, y) for all x, y, s ∈ L. GENERALIZED SYMMETRIC BI-DERIVATIONS OF LATTICES 81 (2) ⇒ (3). Suppose that DT (x, y) ∨ DT (s, y) ≤ DT (x ∨ s, y) and x ≤ s. Then, we get DT (x, y) ≤ DT (x, y)∨DT (s, y) ≤ DT (x∨s, y) = DT (s, y). Therefore, DT is an isotone symmetric bi-(T, F )-derivation on L. (1) ⇒ (3). Suppose DT is an isotone symmetric bi-(T, F )-derivation on L. Since, DT (x, y) ≤ DT (1, y), we have DT (x, y) ≤ F (x) ∧ DT (1, y) by Theorem 3.3 (A). Using Proposition 3.8 and by taking s = 1, we get DT (x, y) = (DT (1, y) ∧ F (x)) ∨ DT (x, y) = DT (1, y) ∧ F (x). (3) ⇒ (4). Assume that DT (x, y) = F (x) ∧ DT (1, y), then DT (x ∧ s, y) = F (x ∧ s) ∧ DT (1, y) = F (x) ∧ F (s) ∧ DT (1, y) = (F (x) ∧ DT (1, y)) ∨ (F (s) ∧ DT (1, y)) = DT (x, y) ∧ DT (s, y) for all x, y, 1 ∈ L (4) ⇒ (1). Let DT (x ∧ s, y) = DT (x, y) ∧ DT (s, y) and x ≤ s. Then, DT (x, y) = DT (x ∧ s, y) = DT (x, y) ∧ DT (s, y). Hence, DT (x, y) ≤ DT (s, y). Theorem 3.10. Let L be a modular lattice and DT be a symmetric bi-(T, F )-derivation on L. Then, followings hold. (1) If DT is an isotone symmetric bi-(T, F )-derivation on L if and only if DT (x ∧ s, y) = DT (x, y) ∧ DT (s, y) (2) If DT is an isotone symmetric bi-(T, F )-derivation on L and F (x ∨ s) = F (x) ∨ F (s), DT (x, y) = F (x), then DT (x ∨ s, y) = DT (x, y) ∨ DT (s, y). Proof. (1) Let DT be a symmetric bi-(T, F )-derivation on L. Since x ∧ s ≤ x and x ∧ s ≤ s, then DT (x ∧ s, y) ≤ DT (x, y) ∧ DT (s, y). Therefore, DT (x, y) ∧ DT (s, y) = = (DT (x, y) ∧ DT (s, y)) ∧ (F (x) ∧ F (s)) (DT (x, y) ∧ F (s)) ∧ (F (x) ∧ DT (s, y)) ≤ (DT (x, y) ∧ F (s)) ∨ (DT (s, y) ∧ F (x)) = DT (x ∧ s, y). Conversely, let DT (x ∧ s, y) = DT (x, y) ∧ DT (s, y) and x ≤ s. Thus, DT (x, y) = DT (x ∧ s, y) = DT (x, y) ∧ DT (s, y), and hence DT (x, y) ≤ DT (s, y) for all x, y, s ∈ L. (2) Let DT be a symmetric bi-T -derivation on L and DT (x, y) = x. Then, by Proposition 3.8 and since L is a modular lattice, thus, DT (s, y) = (DT (s, y) ∨ DT (x ∨ s, y)) ∧ F (s) = F (s) ∧ DT (x ∨ s, y). Thus, DT (x, y) ∨ DT (s, y) = = DT (x, y) ∨ (F (s) ∧ DT (x ∨ s, y)) (DT (x, y) ∨ F (s)) ∧ DT (x ∨ s, y) = (F (x) ∨ F (s)) ∧ DT (x ∨ s, y) = = F (x ∨ s) ∧ DT (x ∨ s, y) DT (x ∨ s, y). Theorem 3.11. Let L be a distributive lattice and DT be a symmetric bi-(T, F )-derivation on L, and F (x ∨ s) = F (x) ∨ F (s). Then, following conditions are hold. (A) If DT is an isotone symmetric bi-(T, F )-derivation on L, then DT (x∧s, y) = DT (x, y)∧ DT (s, y) (B) If DT is an isotone symmetric bi-(T, F )-derivation on L if and only if DT (x ∨ s, y) = DT (x, y) ∨ DT (s, y). 82 CHIRANJIBE JANA AND MADHUMANGAL PAL Proof. Since, DT is an isotone symmetric bi-T -derivation and DT (x ∧ s, y) = DT (x, y) ∧ DT (s, y). By Theorem 3.3 (A), we have DT (x, y) ∧ DT (s, y) = = ((DT (x, y) ∧ F (x)) ∧ ((F (s) ∧ DT (s, y)) (DT (x, y) ∨ F (s)) ∧ (F (x) ∧ DT (s, y) ≤ = (DT (x, y) ∧ F (s)) ∨ (F (x) ∧ DT (s, y) DT (x ∧ s, y). Therefore, DT (x ∧ s, y) = DT (x, y) ∧ DT (s, y) for all x, y, s ∈ L. (B) Let DT be an isotone symmetric bi-(T, F )-derivation. Then, using Theorem 3.3(A) and Proposition 3.8, we have DT (s, y) = (DT (s, y) ∨ (F (s) ∧ DT (x ∨ s, y)) = = (DT (s, y) ∧ F (s)) ∧ (DT (s, y) ∨ DT (x ∨ s, y)) F (s) ∧ DT (x ∨ s, y). In similar way, DT (x, y) = F (x) ∧ DT (x ∨ s, y). Thus, DT (x, y) ∨ DT (s, y) = = (F (x) ∧ DT (x ∨ s, y)) ∨ (F (s) ∧ DT (x ∨ s, y)) (F (x) ∨ F (s)) ∧ DT (x ∨ s, y) = = F (x ∨ s) ∧ DT (x ∨ s, y) DT (x ∨ s, y). Conversely, let DT (x ∨ s, y) = DT (x, y) ∨ DT (s, y) and x ≤ s, then obtained DT (s, y) = DT (x∨s, y) = DT (x, y)∨DT (s, y), which imply DT (x, y) ≤ DT (s, y) for all x, y, s ∈ L. 4. C ONCLUSIONS AND FUTURE WORK In this paper, we discussed the notion of symmetric bi-(T, F )-derivation on lattice and investigated some useful properties of it. In our opinion, these results can be similarly extended to the other algebraic structure such as BCI-algebras, B-algebras, BG-algebras, BF -algebras, M V -algebras, d-algebras, Q-algebras, Incline algebras and so forth. The study of symmetric bi-(T, F )-derivation on different algebraic structures may have a lot of applications in different branches of theoretical physics, engineering, information theory, information retrieval, information control access, cryptanalysis and computer science, etc. We hope that this work will give a deep impact on the upcoming research in this field and other algebraic study to open up a new horizons of interest and innovations. It is our hope that this work would serve as a foundation for further study in the theory of derivations of lattice. 5. 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