Model Companion

A theory T* is a model companion of a theory T when T* is model-complete, T and T* have the same universal consequences, and the models of T* are precisely the existentially closed models of T (equivalently every model of T embeds into a model of T*).

A model companion T* of a theory T is a theory such that (i) T and T* have the same universal consequences, (ii) every model of T embeds into a model of T*, and (iii) T* is model-complete (every embedding between models of T* is elementary). When it exists, T* captures the model-complete closure of the universal part of T.

A theory T* is a model companion of a theory T if T* is model‑complete and every model of T can be embedded into a model of T* (equivalently, T and T* have the same universal consequences), so T* serves as a canonical companion capturing existentially closed behavior relative to T.