A Continuum in the Lattice of Semiring Varieties: The Interval $[\mathsf{V}(S), \mathsf{V}(S^0)]$
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Gao, Zidong, Ren, Miaomiao, Zhao, Xianzhong
Institution: Northwest University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G3090029520, G7299353239
OpenAlex ID: W7213999120