A Reiterman theorem for pseudovarieties of finite first-order structures

We extend Reiterman’s theorem to first-order structures: a class of finite first-order structures is a pseudovariety if and only if it is defined by a set of identities in a certain relatively free profinite structure (pseudoidentities). A well-known result of Birkhoff states that a class of algebras is a variety, that is, is closed under taking subalgebras, homomorphic images and direct products, if and only if it is equational, i.e. it is defined by a set of equations on the corresponding free structures. This result was then extended to firstorder structures [5, 9]: in this framework, varieties are defined by universal positive Horn sentences, i.e. by relational identities (see Section 1.2). Birkhoff’s original statement was generalized in another direction by Reiterman [14]. Reiterman’s theorem states that a class of finite algebras is a pseudovariety (that is, it is closed under taking subalgebras, homomorphic images and finitary direct products) if and only if it is defined by a set of equations in the appropriate free profinite structures. This result has led to a large body of consequences, in particular in finite semigroup theory (see in particular Almeida [1]). ∗Both authors gratefully acknowledge partial support from the following: PRC Mathématique et Informatique, ESPRIT-BRA Working Group 6317 Asmics-2, and Center for Communication and Information Science at the University of Nebraska.