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*).

Principle

A model companion axiomatizes the existentially closed expansions of T: it preserves universal theory while completing existential consequences so that embeddings reflect existential formulas and the companion is model-complete.

Demonstration

The theory of algebraically closed fields of a fixed characteristic is the model companion of the theory of fields of that characteristic: algebraically closed fields are exactly the existentially closed fields, and the companion is model-complete.

Misapplication

Assuming every consistent theory has a model companion; in general a model companion need not exist, and existence can fail for natural classes of structures.

Consequence

When a model companion exists it yields a model-complete theory that often simplifies classification, decidability questions, and transfer of properties because existential closure replaces ad hoc embedding conditions by axioms.

Reversal

The opposite notion is a theory with no companion: many theories admit distinct existentially closed models that cannot be captured by a single model-complete companion, leaving a diverse spectrum of completions.

Boundary

Pertains to pairs of theories (T, T*) where T* shares universal consequences with T; existence is delicate and depends on syntactic and model-theoretic constraints (e.g., consistency of EC-axioms), and the notion is silent about uniqueness up to bi-interpretability beyond standard uniqueness up to logical equivalence.

Semantic Tension

Tension with model completion: a model companion need not be a model completion (which additionally requires that every model of T embeds into a model of the companion and vice versa in a stronger quantifier-eliminating sense); conversely model completions are special model companions.

Synthesis

A model companion is the model-complete theory that captures exactly the existentially closed models of an underlying theory; when it exists it converts existential closure into axioms, enabling a cleaner semantic and syntactic analysis of embeddings and solutions.