## 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.

: Primary 20M05;; secondary 20M20.

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 Thursday, January 14, 2010