From $A$ to $B$ to $Z$

TL;DR

This paper investigates the structure of subvarieties generated by specific semigroups and monoids, showing the existence of a continuum of such subvarieties satisfying particular identities, and demonstrates that certain varieties cannot be finitely defined.

Contribution

It proves the existence of infinitely many subvarieties satisfying specific identities within the varieties generated by ${f A}_2^1$ and ${f B}_2^1$, and shows some varieties are not finitely definable.

Findings

A continuum of subvarieties of ${f A}_2^1$ satisfy $x^2y^2 oughapprox y^2x^2$ and contain ${f B}_2^1$.

The variety of ${f B}_2^1$ cannot be defined within ${f A}_2^1$ by any finite set of identities.

A continuum of subvarieties of ${f B}_2^1$ satisfy $x^2y oughapprox yx^2$ and contain $M({f z}_ty)$.

Abstract

The variety generated by the Brandt semigroup ${\bf B}_2$ can be defined within the variety generated by the semigroup ${\bf A}_2$ by the single identity $x^2y^2\approx y^2x^2$. Edmond Lee asked whether or not the same is true for the monoids ${\bf B}_2^1$ and ${\bf A}_2^1$. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of ${\bf A}_2^1$ that satisfy $x^2y^2\approx y^2x^2$ and contain ${\bf B}_2^1$. A further consequence is that the variety of ${\bf B}_2^1$ cannot be defined within the variety of ${\bf A}_2^1$ by any finite system of identities. Continuing downward, we then turn to subvarieties of ${\bf B}_2^1$. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2y\approx yx^2$ and containing the monoid $M({\bf z}_\infty)$, where ${\bf z}_\infty$ denotes the infinite limit of the Zimin words ${\bf z}_0=x_0$, ${\bf z}_{n+1}={\bf z}_n x_{n+1}{\bf z}_n$.