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Lattices of varieties of plactic-like monoids

Thomas Aird, Duarte Ribeiro

Abstract

We study the equational theories and bases of meets and joins of several varieties of plactic-like monoids. Using those results, we construct sublattices of the lattice of varieties of monoids, generated by said varieties. We calculate the axiomatic ranks of their elements, obtain plactic-like congruences whose corresponding factor monoids generate varieties in the lattice, and determine which varieties are joins of the variety of commutative monoids and a finitely generated variety. We also show that the hyposylvester and metasylvester monoids generate the same variety as the sylvester monoid.

Lattices of varieties of plactic-like monoids

Abstract

We study the equational theories and bases of meets and joins of several varieties of plactic-like monoids. Using those results, we construct sublattices of the lattice of varieties of monoids, generated by said varieties. We calculate the axiomatic ranks of their elements, obtain plactic-like congruences whose corresponding factor monoids generate varieties in the lattice, and determine which varieties are joins of the variety of commutative monoids and a finitely generated variety. We also show that the hyposylvester and metasylvester monoids generate the same variety as the sylvester monoid.
Paper Structure (21 sections, 84 theorems, 43 equations, 3 figures)

This paper contains 21 sections, 84 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Words
  4. Identities and varieties
  5. Properties defining equational theories
  6. Plactic-like monoids
  7. Sublattice of $\mathbb{MON}$ generated by $\mathbf{V}_{\mathsf{sylv}^{\#}}$, $\mathbf{V}_{\mathsf{sylv}}$, $\mathbf{V}_{\mathsf{lSt}}$ and $\mathbf{V}_{\mathsf{rSt}}$.
  8. Varietal meets
  9. Varietal joins
  10. Proving Theorem \ref{['theorem:lattice_1']}
  11. Axiomatic ranks
  12. Sublattice of $\mathbb{MON}$ generated by $\mathbf{V}_{\mathsf{sylv}^{\#}}$, $\mathbf{V}_{\mathsf{sylv}}$, $\mathbf{V}_{\mathsf{lSt}}$, $\mathbf{V}_{\mathsf{rSt}}$ and $\mathbf{V}_{\mathsf{hypo}}$.
  13. Varietal meets and joins
  14. Proving Theorem \ref{['theorem:lattice_2']}
  15. Axiomatic ranks
  16. ...and 6 more sections

Key Result

Lemma 2.1

The equational theory of $\mathbf{V}_{J^1}$ is the set of identities that satisfy id_simple_support_suffix, and the equational theory of $\mathbf{V}_{\overleftarrow{J^1}}$ is the set of identities that satisfy id_simple_support_prefix.

Figures (3)

  • Figure 1: Sublattice $\mathbb{L}_1$ of $\mathbb{MON}$ generated by the varieties generated by the #-sylvester, sylvester, and left and right stalactic monoids.
  • Figure 2: Sublattice $\mathbb{L}_2$ of $\mathbb{MON}$ generated by the varieties respectively generated by the #-sylvester, sylvester, left and right stalactic, and hypoplactic monoids. The points in red correspond to the elements that are not in \ref{['fig:lattice_with_sylv_stal']}.
  • Figure 3: Sublattice $\mathbb{L}_3$ of $\mathbb{MON}$ generated by the varieties respectively generated by the #-sylvester, sylvester, left and right stalactic, and hypoplactic monoids, and the variety $\mathbf{M}_\mathrm{2}$. The points in blue correspond to the elements that are not in \ref{['fig:lattice_with_sylv_stal_hypo']}.

Theorems & Definitions (130)

  • Lemma 2.1
  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4: cain_johnson_kambites_malheiro_representations_2022
  • Theorem 2.5
  • Corollary 2.6: cain_malheiro_ribeiro_sylvester_baxter_2023
  • Theorem 2.7
  • Proposition 2.8: cain_johnson_kambites_malheiro_representations_2022
  • Theorem 2.9
  • Corollary 2.10: cain_malheiro_ribeiro_sylvester_baxter_2023
  • ...and 120 more