What is Feferman–Vaught theorem?

Feferman–Vaught theorem 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 first-order structures to the first-order theory of elements of the structure.

The theorem is considered as one of the standard results in model theory. The theorem extends the previous result of Andrzej Mostowski on direct products of theories.It generalizes the property in universal algebra that equalities carry over to direct products of algebraic structures.