On products of 3-paths in finite full transformation semigroups

Abstract

Let $\operatorname{Sing}_{n}$ denotes the semigroup of all singular self-maps of a finite set $X_n=\lbrace 1, 2,\ldots, n\rbrace$. A map $\alpha\in \operatorname{Sing}_{n}$ is called a $3$-path if there are $i, j, k\in X_n$ such that $i\alpha=j$, $j\alpha=k$ and $x\alpha = x$ for all $x\in X_n\setminus \lbrace i,j\rbrace$. In this paper, we described a procedure to factorise each $\alpha\in \operatorname{Sing}_{n}$ into a product of $3$-paths. The length of each factorisation, that is the number of factors in each factorisation, is obtained to be equal to $\lceil\frac{1}{2}(g(\alpha)+m(\alpha))\rceil$, where $g(\alpha)$ is known as the gravity of $\alpha$ and $m(\alpha)$ is a parameter introduced in this work and referred to as the measure of $\alpha$. Moreover, we showed that $\operatorname{Sing}_n\subseteq P^{[n-1]}$, where $P$ denotes the set of all $3$-paths in $\operatorname{Sing}_n$ and $P^{[k]}=P\cup P^2\cup \cdots \cup P^k$.