Consider a first-order logic signature L.
The definition of product structures takes a family of L-structures for for some index set I and defines the product structure
, which is also an L-structure, with all functions and relations defined pointwise.
The definition generalizes the direct product in universal algebra to structures for languages that contain not only function symbols but also relation symbols.
If is a relation symbol with arguments in L and are elements of the cartesian product, we define the interpretation of in by
When is a functional relation, this definition reduces to the definition of direct product in universal algebra.
For a first-order formula in signature L with free variables, and for an interpretation of the variables , we define the set of indices for which holds in
Given a first-order formula with free variables , there is an algorithm to compute its equivalent game normal form, which is a finite disjunction of mutually contradictory formulas.
The Feferman–Vaught theorem gives an algorithm that takes a first-order formula and constructs a formula that reduces the condition that holds in the product to the condition that holds in the interpretation of sets of indices:
Formula is thus a formula with free set variables, for example, in the first-order theory of Boolean algebra of sets.
It is often of interest to consider substructure of the direct product structure. If the restriction that defines product elements that belong to the substructure can be expressed as a condition on the sets of index elements, then the results can be generalized.
An example is the substructure of product elements that are constant at all but finitely many indices. Assume that the language L contains a constant symbol and consider the substructure containing only those product elements for which the set
is finite. The theorem then reduces the truth value in such substructure to a formula in the boolean algebra of sets, where certain sets are restricted to be finite.
One way to define generalized products is to consider those
substructures where the sets belong to some boolean algebra of sets of indices (a subset of the powerset set algebra ), and where the product substructure admits gluing.[6] Here admitting gluing refers to the following closure condition: if are two product elements and is the element of the boolean algebra, then so is the element defined by "gluing" and according to :
Mostowski, Andrzej (March 1952). "On direct products of theories". Journal of Symbolic Logic. 17 (1): 1–31. doi:10.2307/2267454. Hodges, Wilfrid (1993). "Section 9.6: Feferman-Vaught theorem". Model theory. Cambridge University Press. p. 459. ISBN 0521304423.