Potential isomorphism
In mathematical logic and in particular in model theory, a potential isomorphism is a collection of finite partial isomorphisms between two models which satisfies certain closure conditions. Existence of a partial isomorphism entails elementary equivalence, however the converse is not generally true, but it holds for ω-saturated models.
Definition
A potential isomorphism between two models M and N is a non-empty collection F of finite partial isomorphisms between M and N which satisfy the following two properties:- for all finite partial isomorphisms Z ∈ F and for all x ∈ M there is a y ∈ N such that Z ∪ ∈ F
- for all finite partial isomorphisms Z ∈ F and for all y ∈ N there is a x ∈ M such that Z ∪ ∈ F