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 \(P^{[k]}=P\cup P^2\cup \cdots \cup P^k\).