Minimal relative generating sets of some partial transformation semigroups
Ebru Yiğit, Gonca Ayık, Hayrullah Ayık · Communications in Algebra · 2016
Let PTn,r=Sn∪PKn,r, where Sn is the symmetric group on Xn and PKn,r is the semigroup of partial maps α:Xn→Xn such that |im(α)| ≤ r for 1 ≤ r ≤ n − 1. In this paper, we find necessary and sufficient 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 ∑s=0n−rpr(n−s) where pr(n − s) is the number of partitions of n − s with r terms. We also consider PAn,r=An∪PKn,r, where An is the alternating group on Xn.