Generators and ranks in finite partial transformation semigroups

Abstract

We extend the concept of path-cycles, defined in [2], 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, by analogy with [3], we give a definition for the (m, r)-rank of Pn \ Sn and show that it is n(n+1) 2.