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

Abstract

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