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 \(X_{n}=\{1,2,\ldots,n\}\), 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 \(\frac{n(n+1)}{2}\).