Generators and ranks in finite partial transformation semigroups

We extend the concept of path-cycles, defined in [2], 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, by analogy with [3], we give a definition for the ($m,r$)-rank of $\mathcal{P}_{n}\setminus\mathcal{S}_{n}$ and show that it is $\frac{n(n+1)}{2}$.