On (2; 2)-regular Non-associative Ordered Semigroups via Its Semilattices and Generated (Generalized Fuzzy) Ideals

Yousef Al-Qudah,Mohammad Almousa,Mohammed M Khalaf,Faisal Yousafzai

doi:10.13189/ms.2020.080315

Abstract

The main motivation behind this paper is to study some structural properties of a non-associative structure as it hasn't attracted much attention compared to associative structures. In this paper, we introduce the concept of an ordered A*G**-groupoid and provide that this class is more generalized than an ordered AG-groupoid with left identity. We also define the generated left (right) ideals in an ordered A*G**-groupoid and characterize a (2; 2)-regular ordered A*G**-groupoid in terms of these ideals. We then study the structural properties of an ordered A*G**-groupoid in terms of its semilattices, (2; 2)-regular class and generated commutative monoids. Subsequently, compare -fuzzy left/right ideals of an ordered AG-groupoid and respective examples are provided. Relations between an -fuzzy idempotent subsets of an ordered A*G**-groupoid and its -fuzzybi-ideals are discussed. As an application of our results, we get characterizations of (2; 2)-regular ordered A*G**-groupoid in terms of semilattices and -fuzzy left (right) ideals. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.

Highlights

An AG-groupoid can be referred to as a non-associative semigroup, as the main difference between semigroups and AG-groupoids is the switching of an associative law
The concept of fuzzy sets was first proposed by Zadeh [9] 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
Murali [13] defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset

Summary

Introduction

An AG-groupoid can be referred to as a non-associative semigroup, as the main difference between semigroups and AG-groupoids is the switching of an associative law. A fuzzy subset f of an ordered AG-groupoid S is called an (∈γ, ∈γ ∨qδ)-fuzzy left (right) ideal of S if for all a, b ∈ S and γ, δ ∈ [0, 1], the following conditions hold:. Let f be a fuzzy subset of an ordered AGgroupoid S and γ, δ ∈ [0, 1], f is an (∈γ, ∈γ ∨qδ)-fuzzy left (right, bi-) ideal of S if and only if f satisfies the following conditions.

Basic results and examples

Characterization Problems

Conclusions and Applications