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Finite bases for matrix semirings: transfer theorems and nilpotent counterexamples

  • Jun Jiao
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Abstract

We first show that the nonfinitely based six-element ai-semiring SR6 has finitely based full matrix semirings for every n≥2, with a four-identity basis relative to the ai-semiring axioms. To the best of our knowledge, this is the first known example of this phenomenon. We then construct a countably infinite flat ai-semiring P whose scalar and two-by-two matrix varieties are nonfinitely based, whereas its matrix varieties in sizes n≥3 admit a five-identity relative basis. For both examples, the intervals from the scalar variety to each matrix variety of size at least two have cardinality 2ℵ0. Finally, we establish transfer results for stronger forms of nonfinite basability and a finite-basis dichotomy for matrix semirings of size at least two over scalars with a genuine zero generating a locally finite variety: finite basability in the multiplicatively nilpotent case, and inherent nonfinite basability otherwise.

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Cite

BibTeXRIS
BibTeX
@misc{jiao2026finite,
  title = {{Finite bases for matrix semirings: transfer theorems and nilpotent counterexamples}},
  author = {Jun Jiao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.14551},
  eprint = {2611.14551},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.14551},
}
RIS
TY  - UNPB
TI  - Finite bases for matrix semirings: transfer theorems and nilpotent counterexamples
AU  - Jun Jiao
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.14551
UR  - https://arxiv.org/abs/2611.14551
AB  - We first show that the nonfinitely based six-element ai-semiring SR_6 has finitely based full matrix semirings for every n≥2, with a four-identity basis relative to the ai-semiring axioms. To the best of our knowledge, this is the first known example of this phenomenon. We then construct a countably infinite flat ai-semiring P whose scalar and two-by-two matrix varieties are nonfinitely based, whereas its matrix varieties in sizes n≥3 admit a five-identity relative basis. For both examples, the intervals from the scalar variety to each matrix variety of size at least two have cardinality 2^ℵ_0. Finally, we establish transfer results for stronger forms of nonfinite basability and a finite-basis dichotomy for matrix semirings of size at least two over scalars with a genuine zero generating a locally finite variety: finite basability in the multiplicatively nilpotent case, and inherent nonfinite basability otherwise.
ER  -