A Continuum in the Lattice of Semiring Varieties: The Interval $[\mathsf{V}(S),\mathsf{V}(S^0)]$

Abstract

For an additively idempotent semiring (ai-semiring) $S$, let $S^0$ denote the ai-semiring obtained from $S$ by adjoining a new element $0$. In this paper, we develop an approach to investigate the interval $[\mathsf{V}(S),\mathsf{V}(S^0)]$ of ai-semiring varieties between the variety generated by $S$ and that generated by $S^0$. We establish a general sufficient condition under which this interval has the cardinality of the continuum. This is applied in particular to $[\mathsf{V}(S_7),\mathsf{V}(S_7^0)]$, where $S_7$ is a $3$-element ai-semiring and is a nonfinitely based algebra of the smallest possible order, thereby resolving an open problem proposed by Jackson, Ren, and Zhao (J. Algebra \textbf{611} (2022), 211--245). The same conclusion holds for $[\mathsf{V}(B_2^1),\mathsf{V}((B_2^1{)}^0)]$, where $B_2^1$ is the ai-semiring whose multiplicative reduct is the $6$-element Brandt semigroup. We also present a sufficient condition for the nonfinite basis property in ai-semiring varieties. As a corollary, we obtain a new proof of Dolinka's theorem (Internat. J. Algebra Comput. \textbf{17} (2007), no.~8, 1537--1551) that the $7$-element ai-semiring $(B_2^1{)}^0$ has no finite basis for its identities.