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 free monoids in the correspondence, whereas Pin studied positive varieties of languages and varieties of ordered semigroups. Concerning trees and algebras, similar correspondences were established by Steinby, Almeida, Ésik. The authors have been concerned in several their papers with involving varieties of automata, i.e., unary algebras, in these correspondences. It is well-known that structures of automata, or unary algebras, and their transition semigroups are closely related. Using this, the authors established in [2] a correspondence between regular varieties of unary algebras, suitable classes of semigroups, called k-varieties, and the corresponding congruences on free semigroups. Since the notion of transition semigroup of an algebra treats only regular identities satisfied on it, these semigroups do not contain enough information about algebras satisfying irregular identities. Therefore a new concept for characteristic semigroups of irregular unary algebras, i.e., of directable automata, was introduced and studied in [1]. Moreover, using these semigroups, the authors gave in [1] a correspondence between irregular varieties of unary algebras and corresponding varieties of semigroups. Congruences corresponding to them are studied here and a connection between all these concepts and correspondences is established.