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

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 ${\bf z}_0=x_0$, ${\bf z}_{n+1}={\bf z}_n x_{n+1}{\bf z}_n$.