imtiaz ahmad

We introduce left, right and bi-commutative AG-groupoids. We provide a method to test an arbitrary table of AG-groupoid for these AG-groupoids and explore some of their general properties. Further we study some properties of ideals in these AG-groupoids and decompose left commutative AG-groupoids by introducing some congruences on these AG-groupoids.

In this article we investigate some basic properties of newly discovered classes of AG-groupoid. We consider three classes that include 𝑇1 , 𝑇2 and 𝑇4 -AG-groupoids. We prove that every 𝑇4-AG-groupoid is Bol*-AG-groupoid. We further investigate that 𝑇1 and 𝑇4-AG-groupoids are paramedial and hence are left nuclear square AG-groupoids. We also prove that 𝑇1 and 𝑇4 are transitively commutative AG-groupoids and 𝑇1-AG-3-band is a semigroup.

Quasi-cancellativity is the generalization of cancellativity. We introduce the notion of quasi-cancellativity of semigroup into AG-groupoids. We prove that every AG-band is quasi-cancellative. We also prove the famous Burmistrovich Theorem for AG-groupoids that states “An AG**-groupoid S is a quasi-cancellative if and only if S is a semilattice of cancellative AG**-groupoids”.

A groupoid G is called an AG-groupoid if it satisfies the left invertive law: (ab)c = (cb)a. An AG-group G, is an AG-groupoid with left identity e \in G (that is, ea = a for all a \in G) and for all a \in G there exists a' \in G such that a.a' = a'.a = e. In this article we introduce the concept of AG-groupoids (mod n) and AG-group (mod n) using Vasantha's constructions [1]. This enables us to prove that AG-groupoids (mod n) and AG-groups (mod n) exist for every integer n \geq 3. We also give some nice characterizations of some classes of AG-groupoids in terms of AG-groupoids (mod n).

We show that various classes of algebraic structures such as parame-dial groupoid and AG-groupoid, AG-groupoid and commutative medial, paramedial groupoid and medial, paramedial AG-groupoid and com-mutative semigroup, AG∗∗-groupoid and commutative semigroup are obtainable from each other.

In this paper, we develop the theory of fractional order hybrid differential equations involving Riemann–Liouville differential operators of order [Formula: see text]. We study the existence theory to a class of boundary value problems for fractional order hybrid differential equations. The sum of three operators is used to prove the key results for a couple of hybrid fixed point theorems. We obtain sufficient conditions for the existence and uniqueness of positive solutions. Moreover, examples are also presented to show the significance of the results.

A magma S that meets the identity, xy·z = zy·x, ∀x, y, z ∈ S is called an AG-groupoid. An AG-groupoid S gratifying the paramedial law: uv · wx = xv · wu, ∀ u, v, w, x ∈ S is called a paramedial AGgroupoid. Every AG-grouoid with a left identity is paramedial. We extend the concept of inverse AG-groupoid [4, 7] to paramedial AG-groupoid and investigate various of its properties. We prove that inverses of elements in an inverse paramedial AG-groupoid are unique. Further, we initiate and investigate the notions of congruences, partial order and compatible partial orders for inverse paramedial AG-groupoid and strengthen this idea further to a completely inverse paramedial AG-groupoid. Furthermore, we introduce and characterize some congruences on completely inverse paramedial AG-groupoids and introduce and characterize the concept of separative and completely separative ordered, normal sub-groupoid, pseudo normal congruence pair, and normal congruence pair for the class of completely inv...

In this paper, we define soft union AG-group (abbreviated as soft uni-AG-group).We also define e-set and a-inclusion of soft uni-AG-groups, normal soft uni-AG-subgroups, conjugate of soft uni-AG-groups and commutators of AG-groups. We investigate various properties of these notions and provide a variety of relevant examples that are produced by a computer package GAP to illustrate these notations completely.

We prove some new results for AG*-groupoid such as: (1) an AG*-groupoid is a Bol*-AG-groupoid, (2) an AG*-groupoid is nuclear square AG-groupoid, (3) a cancellative AG*-groupoid is transitively commutative AG-groupoid, (4) a ��-AG-3-band is AG*-groupoid, (5) an AG-groupoid with a left cancellative element is a -AG-groupoid, and (6) an AG*-groupoid is left alternative AG-groupoid.

In this paper we extend the concept of fuzzy AG-subgroups. We introduce some results in normal fuzzy AG-subgroups. We define fuzzy cosets and quotient fuzzy AG-subgroups, and prove that the sets of their collection form an AG-subgroup and fuzzy AG-subgroup respectively. We also introduce the fuzzy Lagrange's Theorem of AG-subgroup. It is known that the condition $\mu(xy)=\mu(yx)$ holds for all $x,y$ in fuzzy subgroups if $\mu$ is normal, but in fuzzy AG-subgroup we show that it holds without normality.

In this paper we introduce the left (right) equal-height elements of a fuzzy power set. We show that both left and right equal-height elements coincide in fuzzy AG-subgroups. We investigate that the collection of left (right) equal-height elements of AG-group G form an AG-subgroup of G. We also establish a relation between the left equal-height elements and left cosets as well as the right equal-height elements and right cosets of an AG-group G.

ABSTRACT An AG-group is a generalization of an abelian group. A groupoid (G, ·) is called an AG-group, if it satisfies the identity (ab)c = (cb)a, called the left invertive law, contains a unique left identity and inverse of its every element. We extend the concept of AG-group to fuzzy AG-group. We define and investigate some structural properties of fuzzy AG-subgroup.

A groupoid with the left invertive law is an LA-semigroup or an Abel-Grassmann’s groupoid (AG-groupoid). This in general is a nonassociative structure that lies between a groupoid and a commutative semigroup. In this note, the significance of the left Abelian distributivie (LAD) LA-semigroup is considered and investigated as a subclass. Various relations with some other known subclasses are established and explored. A hard level problem suggested for LAD-LA-semigroup to be self-dual [29] is solved. Moreover, the notion of ideals is introduced and characterized for the subclass. Several examples and counterexamples generated with the modern tools of Mace-4 and GAP are produced to improve the authenticity of investigated results. AMS (MOS) Subject Classification Codes: 20N02; 20N99