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Cross varieties of aperiodic monoids with commuting idempotents

S. V. Gusev

TL;DR

The paper classifies Cross subvarieties within the class $\mathbf{A}_{\mathsf{com}}$ of aperiodic monoids with commuting idempotents by identifying nine almost Cross subvarieties that obstruct Crossness. It develops a detailed framework of preliminaries on locally finite varieties, word decompositions, and key almost Cross examples ($\mathbf{F},\mathbf{Q},\mathbf{R},\mathbf{P},\mathbf{O}$), and then proves that a subvariety is Cross precisely when it excludes these nine obstructions. The main result shows that the Cross subvarieties form a clean boundary inside $\mathbf{A}_{\mathsf{com}}$, with the obstruction list consisting of $\mathbf{J},\overleftarrow{\mathbf{J}},\mathbf{K},\overleftarrow{\mathbf{K}},\mathbf{L},\mathbf{M},\mathbf{N},\mathbf{P},\overleftarrow{\mathbf{P}}$, each having established status as almost Cross. The methods combine structural word analysis, Rees quotients, and finite-basis arguments to derive smallness and finitely generated properties, culminating in a finite characterization of Cross varieties in this setting. This advances understanding of how Crossness interacts with aperiodicity and commuting idempotents in monoid varieties and yields a precise, practically usable classification.

Abstract

A variety of algebras is called Cross if it is finitely based, finitely generated, and has finitely many subvarieties. In present article, we classify all Cross varieties of aperiodic monoids with commuting idempotents.

Cross varieties of aperiodic monoids with commuting idempotents

TL;DR

The paper classifies Cross subvarieties within the class of aperiodic monoids with commuting idempotents by identifying nine almost Cross subvarieties that obstruct Crossness. It develops a detailed framework of preliminaries on locally finite varieties, word decompositions, and key almost Cross examples (), and then proves that a subvariety is Cross precisely when it excludes these nine obstructions. The main result shows that the Cross subvarieties form a clean boundary inside , with the obstruction list consisting of , each having established status as almost Cross. The methods combine structural word analysis, Rees quotients, and finite-basis arguments to derive smallness and finitely generated properties, culminating in a finite characterization of Cross varieties in this setting. This advances understanding of how Crossness interacts with aperiodicity and commuting idempotents in monoid varieties and yields a precise, practically usable classification.

Abstract

A variety of algebras is called Cross if it is finitely based, finitely generated, and has finitely many subvarieties. In present article, we classify all Cross varieties of aperiodic monoids with commuting idempotents.

Paper Structure

This paper contains 12 sections, 14 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally finite varieties
  4. Subvarieties of $\mathbf{A}_\mathsf{com}$
  5. Words, identities, Rees quotients of free monoids
  6. Decompositions of words
  7. The variety $\mathbf{F}$
  8. The variety $\mathbf{Q}$
  9. The variety $\mathbf{R}$ and its subvarieties
  10. The variety $\mathbf{P}$
  11. The variety $\mathbf{O}$
  12. Main result

Key Result

Lemma 2.1

A locally finite variety $\mathbf{V}$ of algebras is finitely generated if and only if there exists no strictly increasing infinite chain $\mathbf{V}_1 \subset \mathbf{V}_2 \subset \mathbf{V}_3 \subset \cdots$ of varieties such that $\mathbf{V}=\bigvee_{i\ge1}\mathbf{V}_i$. Consequently, any locally

Figures (1)

  • Figure 1: The lattice $L(\mathbf{P})$

Theorems & Definitions (24)

  • Lemma 2.1: Jackson-Lee-18
  • Lemma 2.2
  • proof
  • proof
  • proof
  • Lemma 2.7: Jackson-05
  • Lemma 2.8: Gusev-Vernikov-18
  • Lemma 2.9: Gusev-Vernikov-18
  • Lemma 2.10
  • proof
  • ...and 14 more