Generators and ranks in finite partial transformation semigroups - Garba, G.U.; Imam, A.T. (Algebra and Discrete Mathematics, 2017)
We extend the concept of path-cycle, to the semigroup Pn, of all partial maps on Xn={1,2,…,n}, and show that the classical decomposition of permutations into disjoint cycles can be extended to elements of Pn by means of path-cycles. The device is used to obtain information about generating sets for the semigroup Pn\Sn, of all singular partial maps of Xn. Moreover, we give a definition for the (m,r)-rank of Pn\Sn and show that it is n(n+1)/2.