In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

 and in particular in model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, 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

Saturated model

In mathematical logic, and particularly in its subfield model theory, a saturated model M is one which realizes as many complete types as may be "reasonably expected" given its size...

.

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 ∪ {(x,y)} ∈ F
• for all finite partial isomorphisms Z ∈ F and for all y ∈ N there is a x ∈ M such that Z ∪ {(x,y)} ∈ F

A notion of Ehrenfeucht-Fraïssé game is an exact characterisation of elementary equivalence and potential isomorphism can be seen as an approximation of it. Another notion that is similar to potential isomorphism is that of local isomorphism.