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Varieties of monoids with a distributive subvariety lattice

Sergey V. Gusev

TL;DR

The paper completes the classification of distributive subvariety lattices among aperiodic monoids. It provides an explicit equational description: a distributive variety of aperiodic monoids is contained in $\mathbf B$, $\mathbf D_1$–$\mathbf D_{14}$, $\mathbf D_{15}$, $\mathbf P_n$, $\mathbf Q_n$, $\mathbf R_n$ or their duals, amounting to five infinite series and 29 sporadic cases; the analysis hinges on Sapir's $M_\gamma$-framework, exclusion identities, and stability of $\gamma$-classes. The result shows the distributive landscape is countable and locally finite, contrasting with the uncountable variety lattices in the broader monoid setting. This advances understanding of how lattice-theoretic properties interact with equational constraints in the theory of monoid varieties and their subvarieties. The methods combine deep combinatorial word analysis with algebraic constructions (e.g., Rees quotients and $M_\gamma$-constructions) to isolate exactly which distributive structures can occur.

Abstract

A monoid is aperiodic if all its subgroups are trivial. We completely classify all varieties of aperiodic monoids whose subvariety lattice is distributive.

Varieties of monoids with a distributive subvariety lattice

TL;DR

The paper completes the classification of distributive subvariety lattices among aperiodic monoids. It provides an explicit equational description: a distributive variety of aperiodic monoids is contained in , –, , , , or their duals, amounting to five infinite series and 29 sporadic cases; the analysis hinges on Sapir's -framework, exclusion identities, and stability of -classes. The result shows the distributive landscape is countable and locally finite, contrasting with the uncountable variety lattices in the broader monoid setting. This advances understanding of how lattice-theoretic properties interact with equational constraints in the theory of monoid varieties and their subvarieties. The methods combine deep combinatorial word analysis with algebraic constructions (e.g., Rees quotients and -constructions) to isolate exactly which distributive structures can occur.

Abstract

A monoid is aperiodic if all its subgroups are trivial. We completely classify all varieties of aperiodic monoids whose subvariety lattice is distributive.
Paper Structure (11 sections, 18 theorems, 44 equations, 1 figure)

This paper contains 11 sections, 18 theorems, 44 equations, 1 figure.

Table of Contents

  1. Background and overview
  2. Our main result
  3. Preliminaries
  4. Distributive varieties of aperiodic monoids with commuting idempotents
  5. Words, identities, deduction
  6. Rees quotient monoids
  7. Construction of O. Sapir
  8. Two known results
  9. Exclusion identities
  10. Certain varieties with a non-distributive subvariety lattice
  11. Proof of Theorem \ref{['T: main']}

Key Result

Theorem 1.1

A variety of aperiodic monoids is distributive if and only if it is contained in one of the varieties or the dual ones.

Figures (1)

  • Figure 1: The lattice $\mathfrak L(\mathbf M_\gamma(x^+tyy^+x^+)\vee \mathbf M_\gamma(x^+yy^+tx^+))$

Theorems & Definitions (29)

  • Theorem 1.1
  • Proposition 2.1: Gusev-24
  • Proposition 2.2: Birkhoff's Completeness Theorem for Equational Logic; see Almeida-94
  • Lemma 2.3: Jackson-05
  • Lemma 2.4: Sapir-21
  • Lemma 2.5
  • Lemma 2.6: Zhang-Luo-19,Sapir-21
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 19 more