A group-theoretical invariant for elementary equivalence and its role in representations of elementary classes

There is a natural map which assigns to every modelU of typeτ, (U ε St^τ) a groupG (U) in such a way that elementarily equivalent models are mapped into isomorphic groups.G(U) is a subset of a collection whose members are called Fraisse arrows (they are decreasing sequences of sets of partial isomorphisms) and which arise in connection with the Fraisse characterization of elementary equivalence. LetEC _λ ^U be defined as {U εStr ^τ: ℬ ≡U and |ℬ|=λ; thenEG _λ ^U can be faithfully (i.e. 1-1) represented onto G(U) ×π ^*, whereπ ^*, is a collection of partitions over λ∪λ²∪....