A subdirect decomposition of a semigroup of all fuzzy sets in a semigroup

TL;DR

This paper presents a subdirect decomposition of the semigroup of all fuzzy sets over a semigroup, with a specific operation defined by supremum and conjunction, expanding understanding of fuzzy set algebraic structures.

Contribution

It introduces a novel subdirect decomposition for the semigroup of fuzzy sets over any semigroup, detailing the algebraic structure of fuzzy set operations.

Findings

Provides a detailed algebraic structure of fuzzy set semigroups

Defines a new operation combining fuzzy sets via supremum and conjunction

Extends semigroup theory to fuzzy set contexts

Abstract

In this paper we give a subdirect decomposition of semigroups $({\mathfrak F}(S); \circ )$, where $S$ is a semigroup, ${\mathfrak F}(S)$ is the set of all fuzzy sets in $S$, and the operation $\circ$ on ${\mathfrak F}(S)$ is defined by the following way: for $f, g\in {\mathfrak F}(S)$ and $s\in S$, $(f\circ g)(s)=\vee _{{x, y\in S}\atop{s=xy}}(f(x)\wedge g(y))$ if $s\in S^2$, and $(f\circ g)(s)=0$ otherwise.