A separating identity for matrix semirings over $S_7$

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 $n\ge 2$. 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 $n\ge 3$. 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.