Criterion for a formula-definable quasivariety

In this paper, we study classes of models of a first-order language L with a countable signature σ. For a model A, let Th(A) denote the set of all sentences of L that are true in A, called the elementary type of A. The cardinality of the set T of all elementary types of the signature σ does not exceed the continuum. The product of elementary types of models A and B is defined by Th(A) · Th(B) = Th(A × B), where A × B is the Cartesian product of A and B. Infinite products, ultraproducts, and ultrapowers of elementary types with respect to an ultrafilter D are defined analogously. This yields an algebra hT,·i, which is a commutative semigroup with identity and zero. A binary absorption (recognition) relation is introduced in this semigroup. An elementary type N absorbs an elementary type M if N · M = N. This notion leads to the concept of a formula-definable class of models. Formula-definable classes are closed under ultraproducts as well as finite and infinite direct products; they are idempotently formula-definable and axiomatizable. Varieties and quasivarieties are also considered. All varieties form formula-definable classes of models. Examples of a formula-definable class of models and of a class that is not formula-definable are given. An example of a formula-definable quasivariety that is not a variety is presented. It is shown that not all quasivarieties are formula-definable. Criteria are obtained for a quasivariety to be formula-definable and for a formula-definable class of models to be a quasivariety.