Sukhendu Kar
Jadavpur University, Kolkata, India, Department of Mathematics, Faculty Member
Abstract. In this paper, we introduce fuzzy quasi-ideal and fuzzy biideal of a ternary semigroup and study some related properties of these two subsystems of ternary semigroups. We also characterize fuzzy quasi-ideal and fuzzy bi-ideal of... more
Abstract. In this paper, we introduce fuzzy quasi-ideal and fuzzy biideal of a ternary semigroup and study some related properties of these two subsystems of ternary semigroups. We also characterize fuzzy quasi-ideal and fuzzy bi-ideal of a ternary ...
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We consider a power ternary semigroup [Formula: see text] associated with a ternary semigroup [Formula: see text] and study some properties of [Formula: see text] by using the corresponding properties of [Formula: see text]. After that we... more
We consider a power ternary semigroup [Formula: see text] associated with a ternary semigroup [Formula: see text] and study some properties of [Formula: see text] by using the corresponding properties of [Formula: see text]. After that we study the notion of ordered power ternary semigroup and our main aim is to establish some interconnection between the properties of a ternary semigroup [Formula: see text] and the associated ordered ternary semigroup [Formula: see text].
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In this paper, we study the notions of [Formula: see text]-prime and [Formula: see text]-semiprime ideals of semirings, [Formula: see text]-[Formula: see text]-system and [Formula: see text]-[Formula: see text]-system. We produce some... more
In this paper, we study the notions of [Formula: see text]-prime and [Formula: see text]-semiprime ideals of semirings, [Formula: see text]-[Formula: see text]-system and [Formula: see text]-[Formula: see text]-system. We produce some properties and characterizations for [Formula: see text]-prime and [Formula: see text]-semiprime ideals of semirings in terms of [Formula: see text]-[Formula: see text]-system and [Formula: see text]-[Formula: see text]-system, respectively. We initiate the notion of prime semiring and semiprime semiring to set up characterizations for these two ideals of a semiring. The concept of power semiring associated to a semigroup is also studied in this paper.
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Let [Formula: see text] be the set of all nonempty subsets of a ternary semigroup [Formula: see text]. Then [Formula: see text] is a ternary semigroup with respect to the ternary multiplication defined by [Formula: see text] for all... more
Let [Formula: see text] be the set of all nonempty subsets of a ternary semigroup [Formula: see text]. Then [Formula: see text] is a ternary semigroup with respect to the ternary multiplication defined by [Formula: see text] for all [Formula: see text]. If [Formula: see text] and [Formula: see text] are isomorphic ternary semigroups, then the corresponding power ternary semigroups [Formula: see text] and [Formula: see text] are obviously isomorphic. It is quite natural to ask whether the converse is true, i.e. is it true that for any ternary semigroups [Formula: see text] and [Formula: see text], [Formula: see text] implies that [Formula: see text]? If the class [Formula: see text] of algebra has this property, we say that [Formula: see text] is a globally determined class. In this paper, we provide some class of globally determined ternary semigroups. We show that all ternary semigroups are not globally determined but some special classes of ternary semigroups are globally determin...
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Abstract. The interval-valued prime fuzzy ideals (in brevity, the iv. prime fuzzy ideals) of a semigroup have been recently studied by Kar, Sarkar and Shum [18]. As a continued study of i-v fuzzy ideals, we are going to investigate the... more
Abstract. The interval-valued prime fuzzy ideals (in brevity, the iv. prime fuzzy ideals) of a semigroup have been recently studied by Kar, Sarkar and Shum [18]. As a continued study of i-v fuzzy ideals, we are going to investigate the properties of i-v fuzzy quasi-ideal and i-v fuzzy bi-ideal of a semiring and then we characterize the regularity and intra-regularity of a semiring in terms of the above i-v fuzzy ideals.
I n this paper we introduce the notion of interval-valued fuzzy prime ideal of a semiring, interval-valued fuzzy completely prime ideal of a semiring and study some important properties of these two ideals in the context of... more
I n this paper we introduce the notion of interval-valued fuzzy prime ideal of a semiring, interval-valued fuzzy completely prime ideal of a semiring and study some important properties of these two ideals in the context of interval-valued fuzzy algebra. 2010 Mathematics Subject Classifications: 08A72
Research Interests: Mathematics and Semiring
Let R be a prime ring with its Utumi ring of quotients U , C = Z(U) extended centroid of R, F a nonzero generalized derivation of R, L a noncentral Lie ideal of R and k ≥ 2 a fixed integer. Suppose that there exists 0 6= a ∈ R such that... more
Let R be a prime ring with its Utumi ring of quotients U , C = Z(U) extended centroid of R, F a nonzero generalized derivation of R, L a noncentral Lie ideal of R and k ≥ 2 a fixed integer. Suppose that there exists 0 6= a ∈ R such that a[F (un1 ), un2 , . . . , unk ] = 0 for all u ∈ L, where n1, n2, . . . , nk ≥ 1 are fixed integers. Then either there exists λ ∈ C such that F (x) = λx for all x ∈ R, or R satisfies s4, the standard identity in four variables.
Let R be a commutative ring with 1. In [3], we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if aR + bR = eR for some non-zero idempotent e in R. Let G′(R) be the subgraph... more
Let R be a commutative ring with 1. In [3], we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if aR + bR = eR for some non-zero idempotent e in R. Let G′(R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G′(R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G′(R) is ≤ 2. Finally, we show that the graph G′(R) is a line graph of some graph if and only if R is either a regular ring or a local ring.AMS Subject Classification 2020 : 05C25
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The global determinism of a ternary semigroup [Formula: see text] is the set of all nonempty subsets of [Formula: see text], denoted by [Formula: see text] equipped with the naturally defined multiplication. A class [Formula: see text] of... more
The global determinism of a ternary semigroup [Formula: see text] is the set of all nonempty subsets of [Formula: see text], denoted by [Formula: see text] equipped with the naturally defined multiplication. A class [Formula: see text] of ternary semigroups is said to be globally determined if any two members [Formula: see text] and [Formula: see text] of [Formula: see text] with isomorphic globals are themselves isomorphic i.e. [Formula: see text] implies that [Formula: see text] for any two ternary semigroups [Formula: see text] and [Formula: see text] in the class [Formula: see text]. In this paper, we mainly discuss that the class of all ternary semilattices are globally determined.
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We introduce the notions of quasi-ideal and bi-ideal in ternary semirings and study some properties of these two ideals. We also characterize regular ternary semiring in terms of these two subsystems of ternary semirings.
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In this paper, we study some structure spaces of (m,n)-semiring (S,f,1), introducing the classes of n-ary prime k-ideals, n-ary prime full k-ideals, n-ary prime ideals, maximal ideals and strongly irreducible ideals. Considering their... more
In this paper, we study some structure spaces of (m,n)-semiring (S,f,1), introducing the classes of n-ary prime k-ideals, n-ary prime full k-ideals, n-ary prime ideals, maximal ideals and strongly irreducible ideals. Considering their collections, respectively, of an (m,n)-semiring (S,f, 1), we construct the respective topologies on them by means of closure operator defined in terms of intersection and inclusion relation among these ideals of the (m,n)-semiring (S,f,1). The obtained topological spaces are called the structure spaces of (m,n)-semiring (S,f,1). We mainly study several principal topological axioms and properties of those structure spaces of (m,n)-semiring such as separation axioms, compactness and connectedness etc.
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ABSTRACT The notion of k-regularity in a semiring generalizes the notion of a regular ring introduced by J. Von Neumaan [15]. In semiring theory, the notion of k-regularity was extensively studied by Sen and Bhuniya [17]. In this paper,... more
ABSTRACT The notion of k-regularity in a semiring generalizes the notion of a regular ring introduced by J. Von Neumaan [15]. In semiring theory, the notion of k-regularity was extensively studied by Sen and Bhuniya [17]. In this paper, we study the concept of k-regularities of a semiring in fuzzy setting and characterize some k-regularities of semirings by using different fuzzy ideals of semirings. We also introduce the concept of power fuzzy semirings and study the notion of k-regularities of power fuzzy semirings with the help of regularities of semigroups.
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ABSTRACT
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We introduce the notion of the structure space of Γ-semigroups formed by the class of uniformly strongly prime ideals. We also study separation axioms and compactness property in this structure space.
We introduce the notion of the structure space of ternary semirings formed by the class of prime k-ideals. We also study separation axioms and compactness property of this structure space. Finally, we investigate these properties for the... more
We introduce the notion of the structure space of ternary semirings formed by the class of prime k-ideals. We also study separation axioms and compactness property of this structure space. Finally, we investigate these properties for the structure space of the ternary semiring of non-positive integers ℤ 0 - .
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We introduce the notion of semi-regularity in a ternary semiring and study the Jacobson radical of a ternary semiring by using the notion of semi-regularity. We also study the Jacobson semisimple ternary semirings and weak k-regular... more
We introduce the notion of semi-regularity in a ternary semiring and study the Jacobson radical of a ternary semiring by using the notion of semi-regularity. We also study the Jacobson semisimple ternary semirings and weak k-regular ternary semirings.
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We introduce the notions of semiprime ideal and irreducible ideal in a ternary semiring and study their different properties. We also study the regular ternary semirings and characterize the semiprime k-ideals of the ternary semiring ℤ 0... more
We introduce the notions of semiprime ideal and irreducible ideal in a ternary semiring and study their different properties. We also study the regular ternary semirings and characterize the semiprime k-ideals of the ternary semiring ℤ 0 - .