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

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.