Skip to content
DraconiticResearch
Search Sources API
182,460 studies31 sourcesUpdated 26 min ago
PreprintCC BY 4.0arXiv

The strong nonfinite basis property of S7

  • Lili Wang
  • Qingrui Yin
  • Aifa Wang
Open at sourcePDFDOICite

Abstract

We prove that the three-element additively idempotent semiring S7 is strongly nonfinitely based: every finite additively idempotent semiring whose generated variety contains S7 has no finite equational basis. This answers Problem~1.2 in Yue, Ren and Gao's paper Explicit equational bases for the power semirings of S7 affirmatively. The proof combines a rigidity property of set systems with private points and a probabilistic construction derived from semilinear sets of letter counts in finite semigroups. The argument allows arbitrary polynomial substitutions and does not assume commutative multiplication. As consequences, both power semirings of the multiplicative reduct of S7, and both power semirings of every finite group of order at least three, are strongly nonfinitely based. All algebras are considered in the signature (+,·) without nullary operations.

Related

  1. Strong nonfinite basability of the three-element flat semiring S7
    Mengya Yue, Xiaolei Shao
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. A Generalized L2 Division Theorem and the Strong Openness Property of Multiplier Ideal Sheaves
    Masakazu Takakura
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. The Strong Mankiewicz Property for Closed Convex Body in Normed Spaces
    Ruidong Wang, Jianglong Luo, Li Tang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Baire 1 functions and the strong Choquet property
    Ľubica Holá
    arXivPreprintCC BY-NC-ND 4.0Full textGeometry & topologyarXiv
  5. Counterexamples to the Tensorization Property of Strong Data Processing Inequalities for Quantum Divergences
    Yu Cao
    arXivPreprintarXiv licenceFull textMathematical physicsQuantum physics & quantum informationarXiv

Cite

BibTeXRIS
BibTeX
@misc{wang2026strong,
  title = {{The strong nonfinite basis property of $S_7$}},
  author = {Lili Wang and Qingrui Yin and Aifa Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13980},
  eprint = {2610.13980},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13980},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The strong nonfinite basis property of S_7
AU  - Lili Wang
AU  - Qingrui Yin
AU  - Aifa Wang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.13980
UR  - https://arxiv.org/abs/2610.13980
AB  - We prove that the three-element additively idempotent semiring S_7 is strongly nonfinitely based: every finite additively idempotent semiring whose generated variety contains S_7 has no finite equational basis. This answers Problem~1.2 in Yue, Ren and Gao's paper Explicit equational bases for the power semirings of S_7 affirmatively. The proof combines a rigidity property of set systems with private points and a probabilistic construction derived from semilinear sets of letter counts in finite semigroups. The argument allows arbitrary polynomial substitutions and does not assume commutative multiplication. As consequences, both power semirings of the multiplicative reduct of S_7, and both power semirings of every finite group of order at least three, are strongly nonfinitely based. All algebras are considered in the signature (+,·) without nullary operations.
ER  -