A Reiterman theorem for pseudovarieties of finite first-order structures
Birkhoff's Theorem on classes of algebras being varieties was generalized by J. Reiterman as follows: a class of finite algebras is a pseudovariety (that is, it is closed under taking subalgebras, homomorphic images and finitary direct products) iff it is defined by a set of equations in the appropriate free profinite structures. Here, the result is generalized to first-order structures: under certain natural conditions of finiteness and non-emptiness, pseudovarieties of first-order structures are defined by relational identities (pseudoidentities) in some relatively free profinite structures.
scientific article; zbMATH DE number 920798