Relative ranks of some partial transformation semigroups

Abstract

Let Pn , Tn , In , and Sn be the partial transformation semigroup, the (full) transformation semigroup, thesymmetric inverse semigroup, and the symmetric group on Xn = {1, . . . , n}, respectively. For 1 ≤ r ≤ n−1, let PKn,rbe the subsemigroup consisting α ∈ Pn such that |imα| ≤ r and let SPKn,r = PKn,r Tn . In this paper, we firstexamine the subsemigroup In,r = In ∪ PKn,r and we find the necessary and sufficient conditions for any nonemptysubset of PKn,r to be a (minimal) relative generating set of the subsemigroup In,r modulo In . Then we examine thesubsemigroups PIn,r = SIn ∪ PKn,r and SIn,r = SIn ∪ SPKn,r for 1 ≤ r ≤ n − 1 where SIn = In Sn and computetheir relative rank.