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.

Principle

Model completion upgrades companionship by requiring that existential closure be realized by actual substructure inclusion: the completion supplies canonical completions of models of T within which first‑order types are resolved as much as possible consistent with T.

Demonstration

The theory of algebraically closed fields (ACF) is the model completion of the theory of fields: every field embeds as a subfield of an algebraically closed field, and ACF is model‑complete, so ACF is a model completion of the theory of fields.

Misapplication

Asserting that a model companion is automatically a model completion; some companions exist that do not realize embeddings as substructures for every model of the base theory, so one must check the stronger embedding condition rather than assume it.

Consequence

When a model completion exists, it often yields strong structural control such as quantifier elimination (in favorable languages), a clear description of existentially closed models, and canonical ways to extend models; this facilitates classification and decidability analyses.

Reversal

A theory may have a model companion that is not a model completion: embeddings of models of T into models of the companion may not be realized as substructure inclusions, so the companion does not provide canonical substructure completions.

Boundary

Existence of a model completion is not guaranteed and typically depends on syntactic and semantic properties of T; the notion presupposes a fixed language and Tarski semantics and excludes weaker companions that lack the substructure embedding property.

Semantic Tension

The critical tension is between coronation by model‑completeness alone and the extra geometric/algebraic demand that completions produce substructure inclusions; authors sometimes blur the two, so clarity about the embedding form is essential.

Synthesis

A model completion is the strongest form of model companion: a model‑complete theory that realizes every model of the original theory as a substructure of some model of the completion, thereby providing canonical, existentially closed completions of models of T.