PreprintOn the singular part of the partition monoidJames EastAbstractWe 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 idempotentrank 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.
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