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PreprintCC BY 4.0arXiv

A separating identity for matrix semirings over S7

  • Lili Wang
  • Jiewei Zhang
  • Aifa Wang
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Abstract

Jiao and Ren proved that the varieties generated by the matrix semirings Mn(S7) form an ascending chain of nonfinitely based varieties, and asked whether consecutive members are equal for all n≥2. We give a negative answer by exhibiting an identity in five variables that holds in M2(S7) and fails in M3(S7). Consequently, V(M2(S7)) is a proper subvariety of V(Mn(S7)) for every n≥3. The proof uses the cubic-power property established by Jiao and Ren and a description of products equal to the constant matrix [a]n. The identity forces two distinct indices to exhaust the index set in dimension two; an additional index yields an explicit counterexample in dimension three.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026separating,
  title = {{A separating identity for matrix semirings over $S_7$}},
  author = {Lili Wang and Jiewei Zhang and Aifa Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13981},
  eprint = {2610.13981},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13981},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A separating identity for matrix semirings over S_7
AU  - Lili Wang
AU  - Jiewei Zhang
AU  - Aifa Wang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.13981
UR  - https://arxiv.org/abs/2610.13981
AB  - Jiao and Ren proved that the varieties generated by the matrix semirings M_n(S_7) form an ascending chain of nonfinitely based varieties, and asked whether consecutive members are equal for all n≥2. We give a negative answer by exhibiting an identity in five variables that holds in M_2(S_7) and fails in M_3(S_7). Consequently, V(M_2(S_7)) is a proper subvariety of V(M_n(S_7)) for every n≥3. The proof uses the cubic-power property established by Jiao and Ren and a description of products equal to the constant matrix [a]_n. The identity forces two distinct indices to exhaust the index set in dimension two; an additional index yields an explicit counterexample in dimension three.
ER  -