Regularity of the Semigroups of Transformations with a Fixed Point Set

Ronnason Chinram, Winita Yonthanthum

Abstract

For a nonempty set X,  let T(X) and P(X) denote respectively the full transformation semigroup on X and the partial transformation semigroup on X. For a nonempty subset S of X, we define

T_{\mathcal{F}(S)}(X) = {\alpha \in T(X) \mid  x\alpha = x for all  x \in S},

P_{\mathcal{F}(S)}(X) = {\alpha \in P(X) \mid  S \subseteq \dom \alpha  and x\alpha = x  for all  x \in S\}.

It is obvious that T_{\mathcal{F}(S)}(X) and P_{\mathcal{F}(S)}(X) are subsemigroups of T(X) and  P(X), respectvely.  In this paper, we show that T_{\mathcal{F}(S)}(X) is a  regular semigroup but P_{\mathcal{F}(S)}(X) need not be regular. A necessary and sufficient condition for an element of P_{\mathcal{F}(S)}(X) to be  regular is given. Furthermore, we characterize the left regular and right regular elements of the semigroups T_{\mathcal{F}(S)}(X) and  P_{\mathcal{F}(S)}(X) and made use of these results to deduce the left regularity and right regularity of them.