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PreprintCC BY 4.0arXiv

The finite basis problem for varieties generated by all semirings of a fixed order

  • Aifa Wang
  • Lili Wang
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Abstract

Let Avarn and VVn 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 (+,·) without constants. We prove that both varieties are finitely based if and only if n≤2. The five-element case follows from cyclic identities and a structural analysis of products of squares. For n≥6, each variety contains a finite inherently nonfinitely based ai-semiring. We also prove VVn∩AI=Avarn for every n, where AI is the variety of all ai-semirings.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026finite,
  title = {{The finite basis problem for varieties generated by all semirings of a fixed order}},
  author = {Aifa Wang and Lili Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12486},
  eprint = {2610.12486},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12486},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The finite basis problem for varieties generated by all semirings of a fixed order
AU  - Aifa Wang
AU  - Lili Wang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.12486
UR  - https://arxiv.org/abs/2610.12486
AB  - 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 (+,·) without constants. We prove that both varieties are finitely based if and only if n≤2. The five-element case follows from cyclic identities and a structural analysis of products of squares. For n≥6, each variety contains a finite inherently nonfinitely based ai-semiring. We also prove VV_n∩AI=Avar_n for every n, where AI is the variety of all ai-semirings.
ER  -