Limit varieties of aperiodic monoids
Sergey V. Gusev, Olga B. Sapir
TL;DR
This work advances the theory of limit varieties for aperiodic monoids by presenting a new explicit limit variety built from $\mathbb A_0^1$ and the sigma-extensions of $\mathbb E^1$ and its dual, establishing its non-finite-basis nature. It proves a structural dichotomy (a Sorting Lemma) for any aperiodic variety: it is either hereditary finitely based ($HFB$) or it contains one of a fixed list of benchmark limit varieties, with precise identity-constraints. The paper also proves that the join $\mathbb E^1 \vee \overline{\mathbb E^1}$ is $HFB$ and analyzes the interval up to $\mathbb R_3^1$, culminating in $\mathbb Q^1 \vee \mathbb R_3^1$ being $HFB$ and equal to $\mathbb E^1\{\sigma_2\} \vee \overline{\mathbb E^1\{\sigma_2\}} \vee \mathbb R_3^1$. Together, these results sharpen the landscape of limit varieties for aperiodic monoids, constrain where new non-$FB$ phenomena may occur, and illuminate the lattice structure around idempotent- and $J$-trivial subvarieties.
Abstract
A limit variety is a variety that is minimal with respect to being non-finitely based. We present a new limit variety of aperiodic monoid. We also show that if there exists any other limit variety of aperiodic monoids, then it is contained in the joint of the variety $\mathbb B^1$ of all idempotent monoids and certain finitely generated variety $\mathbb E^1$ with $\mathbb B^1 \wedge \mathbb E^1 = \mathbb L_2^1$, where $\mathbb L_2^1$ is the variety of left-zero monoids. Jackson and Lee proved that $\mathbb E^1$ is HFB, that is, its every subvariety is finitely based. We exend this result a step up the classical decomposition $\mathbb B^1=\bigcup_{i \ge 2} \mathbb L^1_i$ by showing that $\mathbb E^1 \vee \overline{\mathbb E^1} \vee \mathbb L^1_3$ is also HFB, where $\overline{\mathbb E^1}$ is the variety dual of $\mathbb E^1$.