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Minimum transformation representations of diagram monoids

Reinis Cirpons, James East, James D. Mitchell

Abstract

We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid $P_n$ has degree $1 + \frac{B(n+2)-B(n+1)+B(n)}2$ for $n\geq2$, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.

Minimum transformation representations of diagram monoids

Abstract

We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid has degree for , where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.

Paper Structure

This paper contains 23 sections, 33 theorems, 198 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semigroups
  4. Regular $*$-semigroups
  5. Diagram monoids
  6. Transformation monoids and representations
  7. Transformation representations and actions
  8. Partial actions and representations
  9. Actions and right congruences
  10. Right congruences from $\mathscr R$-classes
  11. Regular $*$-semigroups and (partial) actions on projections
  12. The partition, partial Brauer, planar partition and Motzkin monoids
  13. Upper bound
  14. Lower bound
  15. The Temperley--Lieb monoid
  16. ...and 8 more sections

Key Result

Lemma 3.2

Any monogenic monoid action is a right congruence action.

Figures (4)

  • Figure 1: Multiplication of partitions $\alpha,\beta\in\mathcal{P}_6$, with the product graph $\Pi(\alpha,\beta)$ in the middle.
  • Figure 2: Submonoids of $\mathcal{P}_n$ (left) and representative elements from each submonoid (right).
  • Figure 3: A planar partition $\alpha\in\mathscr P\mathcal{P}_8$ (black), with its corresponding Temperley--Lieb element ${\widetilde{\alpha}\in\mathcal{T}\!\mathcal{L}_{16}}$ (orange), illustrating the isomorphism $\mathscr P\mathcal{P}_n\to\mathcal{T}\!\mathcal{L}_{2n}$.
  • Figure 4: The congruence lattice $\operatorname{Cong}(M)$, for $M=\mathcal{P}_n$ or $\mathcal{P}\mathcal{B}_n$ (left), and for $M=\mathscr P\mathcal{P}_n$ or $\mathcal{M}_n$ (right).

Theorems & Definitions (67)

  • Lemma 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.7
  • proof
  • Lemma 3.8
  • proof
  • ...and 57 more