Vol. 55, No. 2, 1974

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The back-and-forth isomorphism construction

Dale Wayne Myers

Vol. 55 (1974), No. 2, 521–529

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 AB.

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.

Mathematical Subject Classification 2000
Primary: 02H05
Secondary: 08A05, 18A15
Received: 11 June 1974
Published: 1 December 1974
Dale Wayne Myers