Green's Relations on Semigroups of Transformations Preserving Two Equivalence Relations

Abstract:

      Let ${\cal T}_X $ be the full transformation semigroup on a set $X$. For a non-trivial equivalence $F$ on $X$, let $T_F (X) =\{ f\in {\cal T}_X : \forall \, (x,y)\in F,\,(f(x),f(y))\in F \}.$ Then $T_F(X) $ is a subsemigroup of ${\cal T}_ X $. Let $E$ be another equivalence on $X$ and $T_{FE}(X)=T_F(X)\cap T_E(X)$. In this paper, under the assumption that the two equivalences $F$ and $E$ are comparable and $E\subseteq F$, we describe the regular elements and characterize Green's relations for the semigroup $T_{FE}(X)$.