Normality of marginal inverse subsemigroups in inverse semigroups

For each pair of integers $m\ge 1$ and $n\ge 2$, we construct a finite inverse semigroup $S_{m,n}\subseteq M_m(\mathbb{C})$ and explicitly determine its marginal inverse subsemigroup $M_{x^n}(S_{m,n})$ associated with the word $x^n$. We prove that the three conditions $\sim_{x^n}=\equiv_{x^n}$ on $S_{m,n}$, $M_{x^n}(S_{m,n})\trianglelefteq S_{m,n}$, and $m\le n$ are equivalent. As a consequence, we answer Problem 11 of Araújo, Araújo and Kinyon in [1]. Moreover, for a word $ω$ in any inverse semigroup $S$, we prove that $\sim_ω=\equiv_ω$ if and only if $M_ω(S)\trianglelefteq S$ and $\sim_ω$ is invariant under inversion.