Year: 2026
Type: preprint
Abstract: Let $\Avar_n$ and $\VV_n$ be the varieties generated by all $n$-element additively idempotent semirings (ai-semirings) and all $n$-element semirings with commutative addition, respectively, in the signature $(+,\cdot)$ without constants. We prove that both varieties are finitely based if and only if $n\leq2$. The five-element case follows from cyclic identities and a structural analysis of products of squares. For $n\geq6$, each variety contains a finite inherently nonfinitely based ai-semiring. We also prove $\VV_n\cap\AI=\Avar_n$ for every $n$, where $\AI$ is the variety of all ai-semirings.
Source: arXiv (Cornell University)
Authors: Wang, Aifa, Wang, Lili
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G6489001854, G7299353239
OpenAlex ID: W7221573598