Fractional domination and fractional total domination on Cayley digraphs of transformation semigroups with fixed sets

Abstract

<abstract><p>For a set $ X $ and a nonempty subset $ Y $ of $ X $, denote by $ T(X) $ the full transformation semigroup under the composition whose elements are functions on $ X $. Let $ Fix(X, Y) $ be the subsemigroup of $ T(X) $ containing functions $ \alpha\in T(X) $ in which each element in $ Y $ is a fixed point of $ \alpha $. Moreover, let $ A $ be a nonempty subset of $ Fix(X, Y) $. The Cayley digraph of $ Fix(X, Y) $ with respect to a connection set $ A $ is a digraph with vertex set $ Fix(X, Y) $ and two vertices $ \alpha, \beta $ induce an arc $ (\alpha, \beta) $ if $ \beta = \alpha\lambda $ for some $ \lambda\in A $. In this paper, the concepts of fractional dominating and fractional total dominating functions of those Cayley digraphs were investigated. Furthermore, the fractional domination and fractional total domination numbers were determined.</p></abstract>