The five-element semigroups $A_2$ and $B_2$ are amongst the most well-known critical examples in finite semigroup theory. Their six element monoid counterparts $A_2^1$ and $B_2^1$ are similarly ubiquitous, with $B_2^1$ sometimes known in the universal algebraic community as Perkin’s Semigroup. The semigroup variety $\mathbb{V}(B_2)$ generated by $B_2$ is defined within $\mathbb{V}(A_2)$ by the law $xxyy=yyxx$ and since $\mathbb{V}(B_2^1)$ also satisfies $xxyy$, it is a natural question as to whether $xxyy = yyxx$ defines $\mathbb{V}(B_2^1)$ within $\mathbb{V}(A_2^1)$ as well. We present an encoding of hypergraph homomorphism problems into semigroups built around $B_2$ to show that there is a continuum of counterexamples to this conjecture, which moreover shows that $\mathbb{V}(B_2^1)$ is not finitely axiomatized within the intersection of $\mathbb{V}(A_2^1)$ with the variety $[[ xxyy = yyxx]]$. With significant care, the proof remains true even in the monoid signature.

This represents the first part of a longer alphabetical journey “From $A$ to $B$ to $Z$” that is joint work with Wenting Zhang (Lanzhou University). In the latter stage of the journey, the $Z$ represents the Zimin words, whose syntactic monoid lies continuum many subvarieties below $\mathbb{V}(B_2^1)$.