Formations of Monoids, Congruences, and Formal Languages

Errata to "Formations of Monoids, Congruences, and Formal Languages"

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

We present an extension of Eilenberg’s variety theorem, a wellknown result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the orig...

Eilenberg theorems for many-sorted formations

A theorem of Eilenberg establishes that there exists a bi-jection 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...

Perfect Congruences on a Free Monoid

Perfect congruences on a free monoid X* are characterized in terms of congruences generated by partitions of X U {1}. It is established that the upper semilattice of perfect congruences if V-isomorphic to the upper semilattice of partitions on Xu{1}. A sublattice of the upper semilattice of perfect congruences is proved to be lattice isomorphic to the lattice of partitions on X. Introduction an...

On varieties of semigroups and unary algebras∗†

The elementary result of Variety theory is Eilenberg’s Variety theorem which was motivated by characterizations of several families of string languages by syntactic monoids or semigroups, such as Schützenberger’s theorem connecting star-free languages and aperiodic monoids. Eilenberg’s theorem has been extended in various directions. For example, Thérien involved varieties of congruences on fre...