Minimal relative generating sets of some partial transformation semigroups

Let (Formula presented.), where Sn is the symmetric group on Xn and PKn,r is the semigroup of partial maps ?Xn › Xn such that |im(?)| ? r for 1 ? r ? n - 1. In this paper, we find necessary and su?cient conditions for any subset of PKn,r to be a (minimal) relative generating set of the subsemigroup PTn,r modulo Sn. Then, for each 1 ? r ? n - 1, we show that the smallest number of elements of PKn,r which, together with Sn, generate PTn,r is (Formula presented.) where pr(n - s) is the number of partitions of n - s with r terms. We also consider (Formula presented.), where An is the alternating group on Xn. © 2017, Copyright © Taylor & Francis Group, LLC