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Juan Climent Vidal, Enric Cosme Llópez.
Eilenberg theorems for many-sorted formations.
Houston J. Math., 45(2):321-369, 2019.

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.

2020 Mathematics Subject Classification: 08A68, 08A70, 68Q70

Keywords: Many-sorted algebra, support, many-sorted congruence, sat-
uration, cogenerated congruence, many-sorted (finite) algebra formation, many-sorted
(finite index) congruence formation, many-sorted regular language formation.