ON CONNECTEDNESS AND COMPLETENESS OF CAYLEY DIGRAPHS OF TRANSFORMATION SEMIGROUPS WITH FIXED SETS

Abstract

Let $\text{Fix}(X,Y)$ be a semigroup of full transformations on a set $X$ in which elements in a nonempty subset $Y$ of $X$ are fixed. In this paper, we construct the Cayley digraphs of $\text{Fix}(X,Y)$ and study some structural properties of such digraphs such as the connectedness and the completeness. Further, some prominent results of Cayley digraphs of $\text{Fix}(X,Y)$ relative to minimal idempotents are verified. In addition, the characterization of an equivalence digraph of the Cayley digraph of $\text{Fix}(X,Y)$ is also investigated.