Some results on semigroups of transformations with restricted range

Abstract

Let XX be a non-empty set and YY a non-empty subset of XX. Denote the full transformation semigroup on XX by T(X)T\left(X) and write f(X)={f(x)∣x∈X}f\left(X)=\{f\left(x)| x\in X\} for each f∈T(X)f\in T\left(X). It is well known that T(X,Y)={f∈T(X)∣f(X)⊆Y}T\left(X,Y)=\{f\in T\left(X)| f\left(X)\subseteq Y\} is a subsemigroup of T(X)T\left(X) and RT(X,Y)RT\left(X,Y), the set of all regular elements of T(X,Y)T\left(X,Y), also forms a subsemigroup of T(X,Y)T\left(X,Y). Green’s ∗\ast -relations and Green’s ∼ \sim \hspace{0.08em}-relations (with respect to a non-empty subset UU of the set of idempotents) were introduced by Fountain in 1979 and Lawson in 1991, respectively. In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup T(X,Y)T\left(X,Y). This investigation proves that the semigroup T(X,Y)T\left(X,Y) is always a left Ehresmann semigroup, and RT(X,Y)RT\left(X,Y) is orthodox (resp. completely regular) if and only if YY contains at most two elements.

Keywords

• green’s ∼-relation
• left (resp. right) ehresmann (resp. restriction) semigroup
• orthodox
• completely regular
• 20m20
• 20m10