The variety generated by all semirings of order three is nonfinitely based
Abstract
We prove that the variety generated by all semirings of order three is nonfinitely based, where addition is not required to be commutative and the signature has no constants. The same conclusion holds for the variety generated by all additively idempotent semirings of order three. We establish these conclusions by excluding a uniform bound on the number of variables in an identity basis. Our proof uses identities associated with anchored odd cycles. Three small commutative test semirings isolate a polynomial equivalence class consisting of exactly two polynomials. For a cycle of length n, every first nontrivial deduction between them requires an identity with at least n+1 variables, even under polynomial substitutions. A retraction followed by a band quotient transfers absorption identities to arbitrary addition and identifies the ai-subvariety of the full joint variety with the joint variety of the ai-generators. Validity of the cycle identities follows from a structural analysis of chain and flat addition. An elementary sixth-power lemma for semigroups of order at most three supplies the retraction.
Related
- The variety generated by all additively idempotent semirings of order four
- The finite basis problem for varieties generated by all semirings of a fixed order
- Inherently nonfinitely based additively idempotent semirings
- A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety
- The finite basis problem for additively idempotent semirings of order four, IV
BibTeX
@misc{wang2026variety,
title = {{The variety generated by all semirings of order three is nonfinitely based}},
author = {Aifa Wang and Wenhao Ju and Lili Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15703},
eprint = {2609.15703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15703},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The variety generated by all semirings of order three is nonfinitely based
AU - Aifa Wang
AU - Wenhao Ju
AU - Lili Wang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15703
UR - https://arxiv.org/abs/2609.15703
AB - We prove that the variety generated by all semirings of order three is nonfinitely based, where addition is not required to be commutative and the signature has no constants. The same conclusion holds for the variety generated by all additively idempotent semirings of order three. We establish these conclusions by excluding a uniform bound on the number of variables in an identity basis. Our proof uses identities associated with anchored odd cycles. Three small commutative test semirings isolate a polynomial equivalence class consisting of exactly two polynomials. For a cycle of length n, every first nontrivial deduction between them requires an identity with at least n+1 variables, even under polynomial substitutions. A retraction followed by a band quotient transfers absorption identities to arbitrary addition and identifies the ai-subvariety of the full joint variety with the joint variety of the ai-generators. Validity of the cycle identities follows from a structural analysis of chain and flat addition. An elementary sixth-power lemma for semigroups of order at most three supplies the retraction.
ER -