Model companion ($T^{\ast }$)

A model companion of $T$ is a model-complete theory $T^{\ast }$ in the same language with $T_{\forall }^{\ast }=T_{\forall }$. Equivalently each model of either theory embeds into a model of the other. Alternating these embeddings and using the elementary chain theorem proves uniqueness up to logical equivalence. A model companion need not be a syntactic extension of its original theory.