Affiliations: [a] Military College of Engineering, National University of Sciences and Technology, Islamabad, Pakistan
| [b] Department of Mathematics, College of Science in Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia
| [c]
Higher Institute of Engineering and Technology, King Mariout, Egypt
| [d] Department of IT, Haripur University, Haripur, K.P.K., Pakistan
| [e] Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
| [f] School of Mathematical Sciences, University of Science and Technology of China, Hefei, China
Correspondence:
[*]
Corresponding author. Arsham Borumand Saeid, Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran. Tel.: +98 34 31322490; Fax: +98 34 33201080; E-mail: arsham@uk.ac.ir.
Abstract: The main motivation behind this paper is to study some structural properties of a non-associative structure (LA-semigroup) in terms of fuzzy sets as it hasn’t attracted much attention compared to associative structures. An LA-semigroup can be referred to as a non-associative semigroup, as the main difference between a semigroup and an LA-semigroup is the switching of an associative law. In this paper, we introduce and characterize fuzzy (0, 2)-ideals, fuzzy (0, 2)-bi-ideals and fuzzy (1, 2)-ideals of an LA-semigroup. We give a new and alternate definition for a strongly regular element of a unitary LA-semigroup and show novel conditions under which a strongly regular LA-semigroup becomes an LA**-semigroup. These characterizations do not exist in literature before this work. As an application of our results we get characterizations of a strongly regular LA-semigroup in terms of fuzzy one-sided ideals and fuzzy bi-ideals. Finally we introduce and investigate the properties of fuzzy inner-unitary LA-subsemigroups in an LA**-semigroup. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.
Keywords: ℒ𝒜-semigroup, strongly regular ℒ𝒜-semigroup, left invertive
law and fuzzy ideals
