The Interval $[\mathsf{V}(S_7),\mathsf{V}(B_2^1)]$ of Semiring Varieties Has the Cardinality of the Continuum

Abstract

We prove that the interval $[\mathsf{V}(S_7),\mathsf{V}(B_2^1)]$ in the lattice of additively idempotent semiring (ai-semiring) varieties has the cardinality of the continuum,where $S_7$ is the smallest nonfinitely based ai-semiring (a three-element algebra), and $B_2^1$ is the ai-semiring whose multiplicative reduct is the six-element Brandt monoid.