Variety (universal Algebra) - Pseudovariety of Finite Algebras

Pseudovariety of Finite Algebras

Since varieties are closed under arbitrary cartesian products, all non-trivial varieties contain infinite algebras. It follows that the theory of varieties is of limited use in the study of finite algebras, where one must often apply techniques particular to the finite case. Attempts have been made to develop a finitary analogue of the theory of varieties.

A pseudovariety is usually defined to be a class of algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. Not every author assumes that all algebras on a pseudovariety are finite; if this is the case, one sometimes talks of a variety of finite algebras. For pseudovarieties, there is no general finitary counterpart to Birkhoff's theorem but in many cases the introduction of a more complex notion of equations allows similar results to be derived.

Pseudovarieties are of particular importance in the study of finite semigroups and hence in formal language theory. Eilenberg's theorem, often referred to as the variety theorem describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups.