Enumeration of idempotents in planar diagram monoids

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...

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Bibliographic Details
Published in:Journal of algebra 2019-03, Vol.522, p.351-385
Main Authors: Dolinka, Igor, East, James, Evangelou, Athanasios, FitzGerald, Des, Ham, Nicholas, Hyde, James, Loughlin, Nicholas, Mitchell, James D.
Format: Article
Language:English
Subjects:
Diagram monoids
Enumeration
Idempotents
Jones monoids
Kauffman monoids
Motzkin monoids
Partition monoids
Temperley–Lieb monoids
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Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2018.11.014