Pseudoelementary class

In logic, a pseudoelementary class is a class of structures derived from an elementary class (one definable in first order logic) by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category theory of (the codomain of) a forgetful functor, and in physics of (hypothesized) hidden variable theories purporting to explain quantum mechanics. Elementary classes are (vacuously) pseudoelementary but the converse is not always true; nevertheless pseudoelementary classes share some of the properties of elementary classes such as being closed under ultraproducts.

A pseudoelementary class is a reduct of an elementary class. That is, it is obtained by omitting some of the sorts and relations of the given class.

A quasivariety defined logically as the class of models of a universal Horn theory can equivalently be defined algebraically as a class of structures closed under isomorphisms, subalgebras, and reduced products. Since the notion of reduced product is more intricate than that of direct product, it is sometimes useful to blend the logical and algebraic characterizations in terms of pseudoelementary classes. One such blended definition characterizes a quasivariety as a pseudoelementary class closed under isomorphisms, subalgebras, and direct products (the pseudoelementary property allows "reduced" to be simplified to "direct").

A corollary of this characterization is that one can (nonconstructively) prove the existence of a universal Horn axiomatization of a class by first axiomatizing some expansion of the structure with auxiliary sorts and relations and then showing that the pseudoelementary class obtained by dropping the auxiliary constructs is closed under subalgebras and direct products. This technique works for example 2 because subalgebras and direct products of algebras of binary relations are themselves algebras of binary relations, showing that the class RRA of representable relation algebras is a quasivariety (and a fortiori an elementary class). This short proof is an effective application of abstract nonsense; the stronger result by Tarski that RRA is in fact a variety required more honest toil.

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