Fuzzy semigroups via semigroups

TL;DR

This paper explores the relationship between fuzzy semigroups and classical semigroup theory, establishing a correspondence that allows properties like regularity to be characterized through fuzzy subsystems.

Contribution

It proves a one-to-one correspondence between fuzzy subsemigroups and certain subsemigroups of a product structure, linking fuzzy and classical semigroup properties.

Findings

Established a bijection between fuzzy subsemigroups and subsemigroups of a product semigroup.

Showed that regularity can be characterized via fuzzy ideals similarly to classical theory.

Unified fuzzy and classical semigroup properties through structural correspondences.

Abstract

The theory of fuzzy semigroups is a branch of mathematics that arose in early 90's as an effort to characterize properties of semigroups by the properties of their fuzzy subsystems which include, fuzzy subsemigroups and their alike, fuzzy one (resp. two) sided ideals, fuzzy quasi-ideals, fuzzy bi-ideals etc. To be more precise, a fuzzy subsemigroup of a given semigroup $(S,\cdot)$ is just a $\wedge$-prehomomorphism $f$ of $(S,\cdot)$ to $([0,1],\wedge)$. Variations of this, which correspond to the other before mentioned fuzzy subsystems, can be obtained by imposing certain properties to $f$. It turns out from the work of Kuroki, Mordeson, Malik and that of many of their descendants, that fuzzy subsystems play a similar role to the structure theory of semigroups that play their non fuzzy analogues. The aim of the present paper is to show that this similarity is not coincidental. As a first step to this, we prove that there is a 1-1 correspondence between fuzzy subsemigroups of $S$ and subsemigroups of a certain type of $S\times I$. Restricted to fuzzy one sided ideals, this correspondence identifies the above fuzzy subsystems to their analogues of $S\times I$. Using these identifications, we prove that the characterization of the regularity of semigroups in terms of fuzzy one sided ideals and fuzzy quasi-ideals can be obtained as an implication of the corresponding non fuzzy analogue.