The Feferman–Vaught theorem[1] in model theory is a theorem by Solomon Feferman and Robert Lawson Vaught that shows how to reduce, in an algorithmic way, the first-order theory of a product of structures to the first-order theory of elements of the structure.
The theorem is considered to be one of the standard results in model theory.
[2][3][4] The theorem extends the previous result of Andrzej Mostowski on direct products of theories.
[5] It generalizes (to formulas with arbitrary quantifiers) the property in universal algebra that equalities (identities) carry over to direct products of algebraic structures (which is a consequence of one direction of Birkhoff's theorem).
Consider a first-order logic signature L. The definition of product structures takes a family of L-structures
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.
are elements of the cartesian product, we define the interpretation of
is a functional relation, this definition reduces to the definition of direct product in universal algebra.
in signature L with free variables a subset of
Given a first-order formula with free variables
, there is an algorithm to compute its equivalent game normal form, which is a finite disjunction
The Feferman–Vaught theorem gives an algorithm that takes a first-order formula
holds in the product to the condition that
can be constructed following the structure of the starting formula
is quantifier free then, by definition of direct product above it follows Consequently, we can take
Extending the condition to quantified formulas can be viewed as a form of quantifier elimination, where quantification over product elements
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
The theorem then reduces the truth value in such substructure to a formula
One way to define generalized products is to consider those substructures where the 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
: The Feferman–Vaught theorem implies the decidability of Skolem arithmetic by viewing, via the fundamental theorem of arithmetic, the structure of natural numbers with multiplication as a generalized product (power) of Presburger arithmetic structures.
, we can define a quotient structure on product elements, leading to the theorem of Jerzy Łoś that can be used to construct hyperreal numbers.