Enumeration of idempotents in planar diagram monoids

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

doi:10.1016/j.jalgebra.2018.11.014

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 journal › Article › peer-review

Abstract

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 language	English
Pages (from-to)	351-385
Number of pages	35
Journal	Journal of Algebra
Volume	522
Early online date	26 Nov 2018
DOIs	https://doi.org/10.1016/j.jalgebra.2018.11.014
Publication status	Published - 15 Mar 2019

Keywords

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

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10.1016/j.jalgebra.2018.11.014

Dolinka_2018_JA_Enumeration_AAM
Copyright © 2018 Elsevier Inc. All rights reserved. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jalgebra.2018.11.014
Accepted author manuscript, 711 KBLicence: CC BY-NC-ND

Cite this

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title = "Enumeration of idempotents in planar diagram monoids",

abstract = "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.",

keywords = "Diagram monoids, Partition monoids, Motzkin monoids, Jones monoids, Temperley–Lieb monoids, Kauffman monoids, Idempotents, Enumeration",

author = "Igor Dolinka and James East and Athanasios Evangelou and Des FitzGerald and Nicholas Ham and James Hyde and Nicholas Loughlin and Mitchell, {James D.}",

year = "2019",

month = mar,

day = "15",

doi = "10.1016/j.jalgebra.2018.11.014",

language = "English",

volume = "522",

pages = "351--385",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press/Elsevier ",

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TY - JOUR

T1 - Enumeration of idempotents in planar diagram monoids

AU - Dolinka, Igor

AU - East, James

AU - Evangelou, Athanasios

AU - FitzGerald, Des

AU - Ham, Nicholas

AU - Hyde, James

AU - Loughlin, Nicholas

AU - Mitchell, James D.

PY - 2019/3/15

Y1 - 2019/3/15

N2 - 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.

AB - 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.

KW - Diagram monoids

KW - Partition monoids

KW - Motzkin monoids

KW - Jones monoids

KW - Temperley–Lieb monoids

KW - Kauffman monoids

KW - Idempotents

KW - Enumeration

UR - https://arxiv.org/abs/1507.04838

U2 - 10.1016/j.jalgebra.2018.11.014

DO - 10.1016/j.jalgebra.2018.11.014

M3 - Article

SN - 0021-8693

VL - 522

SP - 351

EP - 385

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Enumeration of idempotents in planar diagram monoids

Abstract

Keywords

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