Abstract
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 interms of fuzzy left ideals, fuzzy right ideals, fuzzy two-sided ideals, fuzzy generalized bi-ideals, fuzzy bi-ideals, fuzzy interior ideals and fuzzy (1,2)-ideals. We have shown that the set of all fuzzy two-sided ideals of a left regular ordered AG-groupoid forms asemilattice structure. We have characterized all the fuzzy ideals of a left regular ordered AG-groupoid. Finally we have characterized a left regular ordered AG-groupoid by their fuzzy left and fuzzy right ideals.
Highlights
The concept of fuzzy sets was first proposed by Zadeh in 1965, which has a wide range of applications in various fields such as computer engineering, artificial intelligence, control engineering, operation research, management science, robotics and many more
An AG-groupoid S with left identity becomes a semigroup under the binary operation ”◦” defined as, if for all x, y ∈ S, there exists a ∈ S such that x ◦ y =y (Protic, 1994)
A ≤ xa2 =(aa) =(xe) = a2y, by using paramedial law and obviously a ≤ a2y = xa2 holds. This shows that the concept of left regular and right regular coincide in ordered AG-groupoids with left identity
Summary
The concept of fuzzy sets was first proposed by Zadeh in 1965, which has a wide range of applications in various fields such as computer engineering, artificial intelligence, control engineering, operation research, management science, robotics and many more. The left identity of an LA-semigroup allow us to introduce the inverses of elements in an AG-groupoid. If an AG-groupoid contains a left identity, by using medial law, we get a(bc) = b(ac), for all a, b, c ∈ S. An AG-groupoid is a non-associative and non-commutative algebraic structure mid way between a groupoid and a commutative semigroup This structure is closely related with a commutative semigroup, because if an AG-groupoid contains a right identity, it becomes a commutative semigroup (Mushtaq, Q., 1978). An AG-groupoid S with left identity becomes a semigroup under the binary operation ”◦” defined as, if for all x, y ∈ S , there exists a ∈ S such that x ◦ y = (xa)y (Protic, 1994). Throughout in this paper S will be considered as an ordered AG-groupoid unless otherwise specified
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