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

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 $x^2y^2\approx y^2x^2$andcontain${\bf B}_2^1$\mathrm{.}Afurtherconsequenceisthatthevarietyof${\bf B}_2^1$cannotbedefinedwithinthevarietyof${\bf A}_2^1$ 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 $M({\mathbf{z}}_{\mathrm{\infty }})$, where ${\mathbf{z}}_{\mathrm{\infty }}$ denotes the infinite limit of the Zimin words ${\bf z}0=x_0$,${\bf z}{n+1}={\bf z}n x{n+1}{\bf z}_n$.