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Explicit equational bases for the power semirings of S7

  • Mengya Yue
  • Miaomiao Ren
  • Zidong Gao
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Abstract

For every semigroup S, the set 𝒫(S) of all subsets of S and the set 𝒫+(S) of all nonempty subsets of S form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of S, respectively. We investigate the finite basis problem for the full and nonempty power semirings 𝒫(S7) and 𝒫+(S7) of the multiplicative reduct of S7, where S7 is the unique nonfinitely based three-element additively idempotent semiring. We provide explicit infinite equational bases for both and prove that they are nonfinitely based. For 𝒫+(S7), we establish a new sufficient condition for an additively idempotent semiring to be nonfinitely based and apply it to obtain the required result. Moreover, we show that the interval [V(𝒫+(S7)), V(𝒫(S7))] in the lattice of additively idempotent semiring varieties has the cardinality of the continuum.

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Cite

BibTeXRIS
BibTeX
@misc{yue2026explicit,
  title = {{Explicit equational bases for the power semirings of $S_7$}},
  author = {Mengya Yue and Miaomiao Ren and Zidong Gao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19957},
  eprint = {2609.19957},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19957},
}
RIS
TY  - UNPB
TI  - Explicit equational bases for the power semirings of S_7
AU  - Mengya Yue
AU  - Miaomiao Ren
AU  - Zidong Gao
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.19957
UR  - https://arxiv.org/abs/2609.19957
AB  - For every semigroup S, the set 𝒫(S) of all subsets of S and the set 𝒫^+(S) of all nonempty subsets of S form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of S, respectively. We investigate the finite basis problem for the full and nonempty power semirings 𝒫(S_7) and 𝒫^+(S_7) of the multiplicative reduct of S_7, where S_7 is the unique nonfinitely based three-element additively idempotent semiring. We provide explicit infinite equational bases for both and prove that they are nonfinitely based. For 𝒫^+(S_7), we establish a new sufficient condition for an additively idempotent semiring to be nonfinitely based and apply it to obtain the required result. Moreover, we show that the interval [V(𝒫^+(S_7)), V(𝒫(S_7))] in the lattice of additively idempotent semiring varieties has the cardinality of the continuum.
ER  -