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

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 $s\in S$, $(f\circ g)(s)={\vee }_{\genfrac{}{}{0ex}{}{x,y\in S}{s=xy}}(f(x)\wedge g(y))$ if $s\in S^2$, and $(f\circ g)(s)=0$ otherwise.

arXiv comment11 pages