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arXiv · 2610.13981

A separating identity for matrix semirings over $S_7$

Lili Wang·Jiewei Zhang·Aifa Wang
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Abstract

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 $n\geq2$. 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 $n\geq3$. 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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Lili Wang, Jiewei Zhang, Aifa Wang. 2026-09-23. A separating identity for matrix semirings over $S_7$. https://arxiv.org/abs/2610.13981

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