# Potential isomorphism

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In [mathematical logic](/source/Mathematical_logic) and in particular in [model theory](/source/Model_theory), a **potential isomorphism** is a collection of finite [partial isomorphisms](/source/Partial_isomorphism) between two models which satisfies certain closure conditions. Existence of a partial isomorphism entails [elementary equivalence](/source/Elementarily_equivalent), however the converse is not generally true, but it holds for [ω-saturated models](/source/Saturated_model).

## 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](/source/Ehrenfeucht-Fra%C3%AFss%C3%A9_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](/source/Local_isomorphism).

## References

- Chang, C.C. & Keisler, H. Jerome (1989). *Model Theory*. third ed. [Elsevier](/source/Elsevier). ISBN 0-7204-0692-7.
- Poizat, Bruno (2000). [*A Course in Model Theory*](https://archive.org/details/courseinmodelthe0000poiz). [Springer](/source/Springer_Science%2BBusiness_Media). ISBN 0-387-98655-3.

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Adapted from the Wikipedia article [Potential isomorphism](https://en.wikipedia.org/wiki/Potential_isomorphism) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Potential_isomorphism?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.