Eilenberg theorems for many-sorted formations
Juan Climent Vidal, Enric Cosme Llópez
TL;DR
This work generalizes Eilenberg's variety theorem to many-sorted algebras by introducing formations of $\\Sigma$-algebras and $\\Sigma$-congruences for a fixed $S$-sorted signature $\\Sigma$ and showing an isomorphism between their algebraic lattices. Under a finiteness condition on free algebras, it further establishes isomorphisms among finite-index congruence formations, finite $\\Sigma$-algebra formations, and regular-language formations, thereby providing a comprehensive, many-sorted version of Eilenberg-type correspondences. The approach is constructive and congruence-based, relying on saturations and translations rather than coalgebraic methods, and it positions the classic Eilenberg variety theorem as a particular instance within a broad, multi-sorted universal-algebra framework. Together, these results unify automata-theoretic and language-theoretic classifications under a single algebraic formalism, with potential implications for multi-sorted computations and structured language analyses.
Abstract
A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after defining, for a fixed set of sorts $S$ and a fixed $S$-sorted signature $Σ$, the concepts of formation of congruences with respect to $Σ$ and of formation of $Σ$-algebras, we prove that the algebraic lattices of all $Σ$-congruence formations and of all $Σ$-algebra formations are isomorphic, which is an Eilenberg's type theorem. Moreover, under a suitable condition on the free $Σ$-algebras and after defining the concepts of formation of congruences of finite index with respect to $Σ$, of formation of finite $Σ$-algebras, and of formation of regular languages with respect to $Σ$, we prove that the algebraic lattices of all $Σ$-finite index congruence formations, of all $Σ$-finite algebra formations, and of all $Σ$-regular language formations are isomorphic, which is also an Eilenberg's type theorem.