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.
