On the singular part of the partition monoid

James East

Abstract

We study the singular part of the partition monoid \(P_n\); that is, the ideal \(P_n - S_n\), where \(S_n\) is the symmetric group. Our main results are presentations in terms of generators and relations, and we also show that \(P_n - S_n\) is idempotent generated, and that its rank and idempotent-rank are both equal to \(\binom{n+1}{2} = \frac{1}{2}n(n+1)\). One of our presentations uses an idempotent generating set of this minimal cardinality.

Keywords: Partition monoids, Transformation semigroups, Symmetric inverse semigroups, Presentations.

AMS Subject Classification: Primary 20M05;; secondary 20M20.