Frequently in model theory and
occasionally elsewhere, a back-and-forth construction is used to show that any two
countable structures satisfying a given relation are isomorphic. Such a construction is
used to show that any two countable dense linear orders without end points are
isomorphic (Cantor), that any two countable reduced p-primary abelian groups with
the same Ulm invariants are isomorphic (see Kaplansky, Infinite Abelian Groups),
and that any two countable elementarily equivalent saturated structures
are isomorphic (Morley and Vaught). The back-and-forth arguments using
these constructions can often be reduced to an application of the following
result: If R is a symmetric relation between countable structures such that (1)
ARB implies A and B satisfy the same atomic sentences and (2) ARB and
a ∈ A implies there is a b ∈ B such that (A,a)R(B,b), then ARB implies
A≅B.
Loosely, the second condition requires that related structures have enough related
expansions by constants. We prove a similar result in which the second condition
requires, loosely, that related structures have enough similar decompositions into
related components. The prototype of our result is a theorem of Vaught’s on Boolean
algebras mentioned in the last section. In order to suitably formalize “decomposition”
we use category theory.
