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 $(\genfrac{}{}{0ex}{}{n+1}{2})=\frac{1}{2}n(n+1)$. 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.