Eilenberg Theorems for Many-Sorted Formations

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, the concepts of formation of congruences with respect to and of formation of algebras, we prove that the algebraic lattices of all congruence formations and of all algebra formations are isomorphic, which is an Eilenberg's type theorem. Moreover, under a suitable condition on the free algebras and after defining the concepts of formation of congruences of finite index with respect to formation of finite algebras, and of formation of regular languages with respect we prove that the algebraic lattices of all nite index congruence formations, of all nite algebra formations, and of all regular language formations are isomorphic, which is also an Eilenberg's type theorem.