From A to B to Z

The variety generated by the Brandt semigroup B2 can be defined within the variety generated by the semigroup A2 by the single identity x2y2≈ y2x2. Edmond Lee asked whether or not the same is true for the monoids B21 and A21. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of A21 that satisfy x2y2≈ y2x2 and contain B21. A further consequence is that the variety of B21 cannot be defined within the variety of A21 by any finite system of identities. Continuing downward, we then turn to subvarieties of B21. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity x2y≈ yx2 and containing the monoid M(z∞) , where z∞ denotes the infinite limit of the Zimin words z= x, zn+1= znxn+1zn.