In this paper we introduce the concept of ideal elements in poe-AG-groupoid and give some characterizations and properties of their ideal elements. So we consider some results concerning ideals in poe-semigroups and investigate them in poe-AG-groupoids. Also, the class of ideal elements of poe-AG-groupoids are studied, certain intrinsic and basic properties of poe-AG-groupoids including: ideal,...

In this paper, we have introduced the concept of fuzzy ordered AG-groupoids which is the generalization of fuzzy ordered semigroups first considered by Kehayopulu and Tsingelis (2002). We have studied some important features of a left regular ordered AG-groupoid in terms of fuzzy left ideals, fuzzy right ideals, fuzzy two-sided ideals, fuzzy generalized bi-ideals, fuzzy bi-ideals, fuzzy interio...

An AG-groupoid is a non-associative algebraic structure mid way between a groupoid and a commutative semigroup. The left identity in an AGgroupoid if exists is unique [9]. An AG-groupoid is non-associative and non-commutative algebraic structure, nevertheless, it posses many interesting properties which we usually find in associative and commutative algebraic structures. An AG-groupoid with rig...

The concepts of ∈-soft sets and q-soft sets in AG-groupoids are introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy interior ideals and soft interior ideals over an AG-groupoid S. Moreover, we study the relation-ships among (∈, ∈ ∨q)fuzzy interior ideals, ( ) , q ∈∈∨ -fuzzy interior ideals, (α; β)-fuzzy interior ideals and s...

In this paper, we introduce transposition regularity into AG-groupoids, and a variety of regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties structures discussed by their decomposition theorems: (1) L1/R1-transposition equivalent to each other, they can be decomposed the union disjoint Abelian subgroups; (2) LR-transposition an example is given illustrate that ...

Abstract: In this study we introduce a new class of a non-associative algebraic structure namely intra-regular Abel Grassmann's groupoid (AG-groupoid in short). We apply generalized fuzzy ideal theory to this class and discuss its related properties. We introduce ) , ( k q ∨ ∈ ∈ -fuzzy semiprime ideals in AG-groupoids and characterize it. Specifically we have characterized intra-regular AG-grou...

In this paper, we introduce the concept of bipolar-valued fuzzification of ordered AG-groupoids and discuss some structural properties of bipolarvalued fuzzy two-sided ideals of an intra-regular ordered AG-groupoid.

We have given the concept of fuzzy ordered AG**-groupoids and studied some important structural properties of a (2,2) regular ordered AG**-groupoid in terms of different fuzzy ideals.

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’s 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”.

In this paper, we introduce the concept of bipolar-valued fuzzification of ordered AG-groupoids and discuss some structural properties of bipolarvalued fuzzy two-sided ideals of an intra-regular ordered AG-groupoid.