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Limit varieties of monoids satisfying a certain identity

Sergey V. Gusev, Yu Xian Li, Wen Ting Zhang

TL;DR

The paper resolves the finite-basis landscape for a key identity in aperiodic monoids by completely classifying limit subvarieties of the base variety $\\mathbf P = var\\{ xsxt \\approx xsxtx \\}$. It introduces the critical-pair framework and a hierarchy of derived identities, establishing two explicit limit subvarieties, $\\mathbf J_1$ and $\\mathbf J_2$, with $\\mathbf J_2$ exhibiting countably many subvarieties. The authors further show that the variety generated by any monoid of order at most five is hereditary finitely based, while larger examples exist with richer subvariety lattices; they also analyze the specific monoid $P_2^1$, proving it is HFB and detailing its subvariety structure. Collectively, the work narrows the search for limit varieties in monoids to the $\\mathbf P$-realm and clarifies how small-order monoids constrain or permit diverse subvariety lattices. This has implications for the finite basis problem and the understanding of the landscape of limit varieties in the aperiodic monoid setting, including the existence of limit varieties with infinitely many subvarieties.

Abstract

A limit variety is a variety that is minimal with respect to being non-finitely based. Since the turn of the millennium, much attention has been given to the classification of limit varieties of aperiodic monoids. Seven explicit examples have so far been found, and the task of locating other examples has recently been reduced to two subproblems, one of which is concerned with monoids that satisfy the identity $xsxt \approx xsxtx$. In the present article, we provide a complete solution to this subproblem by showing that there are precisely two limit varieties that satisfy this identity. One of them turns out to be the first example having infinitely many subvarieties. It is also deduced that the variety generated by any monoid of order five or less contains at most countably many subvarieties.

Limit varieties of monoids satisfying a certain identity

TL;DR

The paper resolves the finite-basis landscape for a key identity in aperiodic monoids by completely classifying limit subvarieties of the base variety . It introduces the critical-pair framework and a hierarchy of derived identities, establishing two explicit limit subvarieties, and , with exhibiting countably many subvarieties. The authors further show that the variety generated by any monoid of order at most five is hereditary finitely based, while larger examples exist with richer subvariety lattices; they also analyze the specific monoid , proving it is HFB and detailing its subvariety structure. Collectively, the work narrows the search for limit varieties in monoids to the -realm and clarifies how small-order monoids constrain or permit diverse subvariety lattices. This has implications for the finite basis problem and the understanding of the landscape of limit varieties in the aperiodic monoid setting, including the existence of limit varieties with infinitely many subvarieties.

Abstract

A limit variety is a variety that is minimal with respect to being non-finitely based. Since the turn of the millennium, much attention has been given to the classification of limit varieties of aperiodic monoids. Seven explicit examples have so far been found, and the task of locating other examples has recently been reduced to two subproblems, one of which is concerned with monoids that satisfy the identity . In the present article, we provide a complete solution to this subproblem by showing that there are precisely two limit varieties that satisfy this identity. One of them turns out to be the first example having infinitely many subvarieties. It is also deduced that the variety generated by any monoid of order five or less contains at most countably many subvarieties.

Paper Structure

This paper contains 17 sections, 53 theorems, 166 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Words, identities, and deduction
  4. $k$-decomposition of a word
  5. Some known facts
  6. Some properties of the variety $\mathbf P$
  7. Critical pairs
  8. Removing critical pairs of the form $\{{_1}x,{_2}y\}$
  9. Removing critical pairs of the form $\{{_2}x,{_2}y\}$
  10. Limit subvarieties of $\mathbf P$
  11. The limit variety $\mathbf J_1$
  12. The limit variety $\mathbf J_2$
  13. Classification of limit subvarieties of $\mathbf P$
  14. The monoid $P_2^1$
  15. The variety $\mathbf P_2^1$ is HFB
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $\mathbf V$ be the variety defined by some set $\Sigma$ of identities. Then $\mathbf V$ satisfies an identity $\mathbf u \approx \mathbf v$ if and only if $\mathbf u \approx \mathbf v$ is deducible from $\Sigma$.∎

Figures (2)

  • Figure 1: The lattice $\mathfrak L(\mathbf P_2^1)$
  • Figure 2: A modular non-distributive lattice

Theorems & Definitions (97)

  • Remark 1.1
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3: Gusev and Vernikov Gusev-Vernikov-18
  • Lemma 2.4: Gusev and Vernikov Gusev-Vernikov-18
  • Lemma 2.5: Gusev and Vernikov Gusev-Vernikov-18
  • Lemma 2.6: Gusev and Vernikov Gusev-Vernikov-18
  • Remark 2.7
  • Corollary 2.8
  • Lemma 2.9
  • ...and 87 more