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

TL;DR

This paper demonstrates that the interval between two specific additively idempotent semiring varieties in the lattice has the cardinality of the continuum, revealing a rich and complex structure.

Contribution

It establishes that the interval between the varieties generated by S_7 and B_2^1 has the cardinality of the continuum, a novel result in semiring variety theory.

Findings

The interval's cardinality is continuum-sized.

S_7 is the smallest nonfinitely based ai-semiring.

B_2^1 is based on the six-element Brandt monoid.

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.