Extended Green Relations and Rank Pair Enumeration in Finite Rhotrix Semigroups

Let $q$ be a prime power and let $T_m(q)$ be the monoid of order-$(2m-1)$ Rhotrix arrays whose row–column multiplication corresponds to componentwise multiplication in $M_m(\mathbb F_q)\times M_{m-1}(\mathbb F_q)$. The extended Green relations $\mathcal L^*$ and $\mathcal R^*$ are characterized by the row and column spaces of both matrix components. Regularity implies $\mathcal L^*=\mathcal L$ and $\mathcal R^*=\mathcal R$, while $\mathcal D^*=\mathcal D=\mathcal J$ is indexed by the pair of component ranks. For each rank pair, closed formulas are derived for the sizes and numbers of $\mathcal L^*$-, $\mathcal R^*$-, and $\mathcal H^*$-classes, the number of idempotents, and the order of every maximal subgroup. The principal ideal order, all two-sided ideals, and a rank-generating polynomial are determined. An exhaustive computation for $T_2(2)$ confirms the formulas and the translation-kernel definitions. These results provide an explicit finite-field structure theory for the specified row–column Rhotrix monoid.