A Subdirect Decomposition of a Semigroup of All Fuzzy Sets in a Semigroup

Abstract

Divisibility of elements in a semigroup and the decomposition of a semigroup into subdirect product of semigroups are two basic concepts in the study of the structure of semigroups. The main object of our present paper is to show that the divisibility of elements in a semigroup $S$ determines a decomposition of a semigroup $({\mathfrak{F}}(S);\circ)$ of all fuzzy sets in $S$ into a subdirect product of semigroups, where the operation $\circ$ on ${\mathfrak{F}}(S)$ is defined by the following way: for fuzzy set $f$ and $g$ in $S$, $(f\circ g)(s)=0$ if $s\in S\setminus S^{2}$, and $(f\circ g)(s)=\bigvee_{{x,y\in S}\atop{s=xy}}(f(x)\bigwedge g(y))$ if $s\in S^{2}$.