A step to compute the determinant of finite semigroups not in ECom
M. H. Shahzamanian
TL;DR
The paper addresses the problem of computing nonzero determinants $\theta_S(X)$ for finite semigroups beyond $\mathsf{ECom}$ by introducing $\ll$-smooth singleton-rich semigroups and proving a determinant factorization over idempotents: $\theta_S(X)=\pm\prod_{e\in E(S)} \widetilde{\theta}_e(Y_e)$ with $Y_e=\{y_s: s\in \widetilde{L}_e\widetilde{R}_e\}$ and $y_s=\sum_{t\ll s}\mu_S(t,s)x_t$, so $\theta_S(X)\neq 0$ iff all $\widetilde{\theta}_e(Y_e)\neq 0$ for every idempotent $e$. It develops the necessary algebraic toolkit (Green’s relations, incidence algebras, Möbius inversion) and introduces a transitive, singleton-rich framework ($\ll$ and $\lll$) to enable factorization and computation, with algorithmic verification and explicit examples (including $S_9$). The results offer a structural route to extend MacWilliams-type theorems for codes over semigroup algebras and provide concrete finite examples and computational methods to test nonzero determinants. Overall, this work lays groundwork for systematic determinant computation outside the $\mathsf{ECom}$ pseudovariety by tying determinant nonvanishing to idempotent-structure via a refined order and factorization.
Abstract
The purpose of this paper is to begin studying the computation of the nonzero determinant of semigroups within the class of finite semigroups that possesses a pair of non-commutative idempotents. This paper focuses on a class of these semigroups introduced as $\ll$-smooth semigroups. This computation is applicable in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.