*Semantics Club 24 11 2000*

# Compositional theorem for generalized sum

### Abstract

Composition Theorems are tools which reduce sentences
about a complex structure to sentences about its parts.
A seminal example of such a result is the Feferman-Vaught Theorem (1959)
which reduces the first-order theory of generalized products
to the first order theory of its factors and
the monadic second-order theory of index structure.
In this talk I explain:

- a definition of a generalized sum of structure and
- a composition theorem for first-order logic over the generalized sum.

Some applications of the composition theorem to definability will be
provided which replace game (or inductive) arguments by transparent
reductions.