Fuzzy semigroups in reductive semigroups

Abstract

We consider a fuzzy semigroup $S$ in a right (or left) reductive semigroup $X$ such that
$S(k)=1$ for some $k \in X$ and find a faithful representation (or anti-representation) of $S$
by transformations of $S$.
Also we show that a fuzzy semigroup $S$ in a weakly reductive semigroup $X$ such that
$S(k)=1$ for some $k \in X$ is isomorphic to the semigroup
consisting of all pairs of inner right and left translations of
$S$ and that $S$ can be embedded into the semigroup consisting of all pairs of linked right and left
translations of $S$ with the property that $S$ is an ideal of the semigroup.