Generators and ranks in finite partial transformation semigroups

Abstract

We extend the concept of path-cycle, to the semigroup ${\mathcal{P}}_n$, of all partial maps on $X_n=\{1,2,\dots ,n\}$, and show that the classical decomposition of permutations into disjoint cycles can be extended to elements of ${\mathcal{P}}_n$ by means of path-cycles. The device is used to obtain information about generating sets for the semigroup ${\mathcal{P}}_n\setminus {\mathcal{S}}_n$, of all singular partial maps of $X_n$. Moreover, we give a definition for the ($m,r$)-rank of ${\mathcal{P}}_n\setminus {\mathcal{S}}_n$ and show that it is $\frac{n(n+1)}{2}$.