 ##  [Model Companion](/model-companion) 

  ##  [Model Companion](https://mathlogic.quantumdictionary.io/model-companion-0) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

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*).

 
 ##  [Model Companion](https://puremath.quantumdictionary.io/model-companion-1) 

  
 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

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.

 
 ##  [Model Companion](https://algebra.quantumdictionary.io/model-companion-2) 

  
 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 
Definition

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.