
A theorem of Eilenberg establishes that there exists a bijectionbetween 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 sortsSand a fixedSsorted signature Σ, the concepts of formation of congruenceswith respect to Σ and of formation of Σalgebras, we prove that the algebraic lattices of all Σcongruence formations and of all Σalgebra formationsare isomorphic, which is an Eilenberg's type theorem. Moreover, under asuitable condition on the free Σalgebras and after defining the concepts offormation of congruences of finite index with respect to Σ, of formation offinite Σalgebras, and of formation of regular languages with respect to Σ, weprove that the algebraic lattices of all Σfinite index congruence formations,of all Σfinite algebra formations, and of all Σregular language formationsare isomorphic, which is also an Eilenberg's type theorem.
