Nonfinitely based intervals of power semiring varieties

Abstract

We study intervals in the lattice of additively idempotent semiring varieties associated with power semirings of finite groups. We prove that every variety lying above the nonempty power semiring of a finite group of order at least three and below a variety generated by finitely many full power semirings of finite groups is nonfinitely based. In fact, none of these varieties admits an identity basis with a fixed finite bound on the number of variables. In particular, adjoining the empty set gives an interval consisting entirely of nonfinitely based varieties. We also construct equational upper bounds that are closed under zero adjunction and yield further intervals with the same property. The proof combines finite commutative quotient semirings, a cardinality estimate for kernel blockers over finite modules, and a uniform estimate for fibres of ordered products in finite groups. As a consequence, the full power semiring of a finite group is finitely based precisely when the group has at most two elements.

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BibTeX RIS

BibTeX

@misc{shao2026nonfinitely,
  title = {{Nonfinitely based intervals of power semiring varieties}},
  author = {Xiaolei Shao and Zidong Gao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26019},
  eprint = {2609.26019},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26019},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Nonfinitely based intervals of power semiring varieties
AU  - Xiaolei Shao
AU  - Zidong Gao
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26019
UR  - https://arxiv.org/abs/2609.26019
AB  - We study intervals in the lattice of additively idempotent semiring varieties associated with power semirings of finite groups. We prove that every variety lying above the nonempty power semiring of a finite group of order at least three and below a variety generated by finitely many full power semirings of finite groups is nonfinitely based. In fact, none of these varieties admits an identity basis with a fixed finite bound on the number of variables. In particular, adjoining the empty set gives an interval consisting entirely of nonfinitely based varieties. We also construct equational upper bounds that are closed under zero adjunction and yield further intervals with the same property. The proof combines finite commutative quotient semirings, a cardinality estimate for kernel blockers over finite modules, and a uniform estimate for fibres of ordered products in finite groups. As a consequence, the full power semiring of a finite group is finitely based precisely when the group has at most two elements.
ER  -

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