Enumeration of idempotents in planar diagram monoids

Igor Dolinka, James East*, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, James D. Mitchell

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
7 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)351-385
Number of pages35
JournalJournal of Algebra
Early online date26 Nov 2018
Publication statusPublished - 15 Mar 2019


  • Diagram monoids
  • Partition monoids
  • Motzkin monoids
  • Jones monoids
  • Temperley–Lieb monoids
  • Kauffman monoids
  • Idempotents
  • Enumeration


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