Model Completion

A refinement of a first-order theory T producing a theory T* (when it exists) whose models are exactly the existentially closed models of T; T* is model-complete if every embedding between models of T* is elementary, equivalently every formula is equivalent to an existential formula in T*.

A model completion of a first-order theory T is a theory T* that is a model companion of T, is model-complete (every embedding between models of T* is elementary), and whose models are precisely the existentially closed models of T.

A model completion of a theory T is a model companion T* with the stronger property that every model of T embeds as a substructure (not merely embeds) into a model of T*, equivalently T* is model‑complete and every model of T has an existentially closed extension that is a superstructure in which the embedding is an inclusion.