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PreprintCC BY 4.0arXiv

Finite bases for full power semirings of finite nilpotent semigroups

  • Lili Wang
  • Qingrui Yin
  • Aifa Wang
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Abstract

We investigate finite equational bases for full power semirings of finite semigroups, including the empty set, in the constant-free signature with addition and multiplication. For a nontrivial finite nilpotent semigroup, we associate a finite relational structure recording the ordered products that are nonzero. We prove that the full power semiring is finitely based if and only if this structure has finite duality, and relate this condition to first-order definability and dismantling of the square of its core. Quantitative bounds connect obstruction size with the number of variables required in an identity basis. In the commutative case of nilpotency index d, the criterion reduces to the existence of an element with nonzero (d−1)st power. We also establish a nonfinite-basis obstruction for semigroups with a two-element group ideal and prove a finite lifting theorem. These results yield a direct-product criterion and a five-element counterexample to sufficiency of the identity-fibre condition. The arguments use equational logic, finite relational duality, and explicit algebraic constructions.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026finite,
  title = {{Finite bases for full power semirings of finite nilpotent semigroups}},
  author = {Lili Wang and Qingrui Yin and Aifa Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01329},
  eprint = {2610.01329},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01329},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finite bases for full power semirings of finite nilpotent semigroups
AU  - Lili Wang
AU  - Qingrui Yin
AU  - Aifa Wang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01329
UR  - https://arxiv.org/abs/2610.01329
AB  - We investigate finite equational bases for full power semirings of finite semigroups, including the empty set, in the constant-free signature with addition and multiplication. For a nontrivial finite nilpotent semigroup, we associate a finite relational structure recording the ordered products that are nonzero. We prove that the full power semiring is finitely based if and only if this structure has finite duality, and relate this condition to first-order definability and dismantling of the square of its core. Quantitative bounds connect obstruction size with the number of variables required in an identity basis. In the commutative case of nilpotency index d, the criterion reduces to the existence of an element with nonzero (d−1)st power. We also establish a nonfinite-basis obstruction for semigroups with a two-element group ideal and prove a finite lifting theorem. These results yield a direct-product criterion and a five-element counterexample to sufficiency of the identity-fibre condition. The arguments use equational logic, finite relational duality, and explicit algebraic constructions.
ER  -