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Varieties of aperiodic monoids with commuting idempotents whose subvariety lattice is distributive

Sergey V. Gusev

TL;DR

The paper achieves a complete classification of distributive subvariety lattices within the class $oldsymbol{A}_{ ext{com}}$ of aperiodic monoids with commuting idempotents. It introduces and leverages Olga Sapir’s construction, along with stability and deduction tools, to identify 14 base varieties $oldsymbol{D}_1,dots,oldsymbol{D}_{14}$ such that every distributive subvariety of $oldsymbol{A}_{ ext{com}}$ is contained in one of them. The authors further show that all distributive subvarieties arise from 5 infinite series and 27 sporadic examples, yielding a countable landscape of distributive cases. The sufficiency part proves each $oldsymbol{D}_i$ is distributive, while the necessity part rules out other non-distributive configurations through a web of separating identities and non-distributive constructions. Overall, the work advances the distinguished program of classifying distributive subvarieties in monoid varieties and provides a concrete, checkable framework for $oldsymbol{A}_{ ext{com}}$.

Abstract

We completely classify all varieties of aperiodic monoids with commuting idempotents whose subvariety lattice is distributive.

Varieties of aperiodic monoids with commuting idempotents whose subvariety lattice is distributive

TL;DR

The paper achieves a complete classification of distributive subvariety lattices within the class of aperiodic monoids with commuting idempotents. It introduces and leverages Olga Sapir’s construction, along with stability and deduction tools, to identify 14 base varieties such that every distributive subvariety of is contained in one of them. The authors further show that all distributive subvarieties arise from 5 infinite series and 27 sporadic examples, yielding a countable landscape of distributive cases. The sufficiency part proves each is distributive, while the necessity part rules out other non-distributive configurations through a web of separating identities and non-distributive constructions. Overall, the work advances the distinguished program of classifying distributive subvarieties in monoid varieties and provides a concrete, checkable framework for .

Abstract

We completely classify all varieties of aperiodic monoids with commuting idempotents whose subvariety lattice is distributive.
Paper Structure (24 sections, 73 theorems, 312 equations, 1 figure, 1 table)

This paper contains 24 sections, 73 theorems, 312 equations, 1 figure, 1 table.

Table of Contents

  1. Background and overview
  2. The main result
  3. Preliminaries
  4. Deduction
  5. Rees quotient monoids
  6. Decomposition of words
  7. Construction of Olga Sapir
  8. Definition
  9. Several certain congruences on $\mathfrak X^\ast$
  10. Defining congruence classes by regular expressions
  11. Lattices of subvarieties of two varieties
  12. Separating identities for varieties
  13. Many certain identities
  14. Identities in $\Phi_1$
  15. Identities in $\Phi_2$
  16. ...and 9 more sections

Key Result

Theorem 1.1

A subvariety of $\mathbf A_\mathsf{com}$ is distributive if and only if it is contained in one of the varieties or the dual ones.

Figures (1)

  • Figure 1: The lattice $\mathfrak L\left(\mathbf M_\lambda(xzyx^+ty^+)\right)$

Theorems & Definitions (126)

  • Theorem 1.1
  • Proposition 2.1: Birkhoff's Completeness Theorem for Equational Logic; see Almeida-94
  • Lemma 2.2: Jackson-05
  • Lemma 2.3: Gusev-Vernikov-21
  • Lemma 2.4
  • Lemma 3.1: Sapir-21
  • Lemma 3.2: Sapir-21
  • Lemma 3.3: Sapir-21
  • Lemma 3.4
  • proof
  • ...and 116 more