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

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 $x^2y^2approx y^2x^2$. 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 $x^2y^2approx y^2x^2$ 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 $x^2yapprox yx^2$ and containing the monoid $M({bf z}_infty)$, where ${bf z}_infty$ denotes the infinite limit of the Zimin words ${bf z}_0=x_0$, ${bf z}_{n+1}={bf z}_n x_{n+1}{bf z}_n$.