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Enumeration of idempotents in planar diagram monoids

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We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex- and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular *-) semigroups, and give several tables of calculated values.

Author(s): Igor Dolinka, James East, Des FitzGerald, Nicholas Ham, James Hyded, Nicholas Loughlin, James D. Mitchell
Publisher: Journal of Algebra
Year: 2019

Language: English
Pages: 351-385
Tags: Diagram monoids, partition monoids, Motzkin monoids, Jones monoids, Temperley-Lieb monoids, Kauffman monoids, idempotents, enumeration. MSC: 05E15, 20M20, 20M17, 05A18.

1 Introduction......Page 1
2 Existing algorithms......Page 2
3.1 Definitions and preliminaries......Page 5
3.2 Interface graphs and characterisation of idempotents......Page 7
3.3 A mapping on E(Mn) and an enumeration method......Page 9
4.1 Background on Dyck and Motzkin words......Page 16
4.2 The algorithm for Jones idempotents......Page 18
4.4 The algorithm for Motzkin idempotents......Page 19
5 Values and benchmarking......Page 20