##  [Elementary Extension](/elementary-extension) 

  ##  [Elementary Extension](https://mathlogic.quantumdictionary.io/elementary-extension-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 & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A structure N is an elementary extension of a structure M (same signature) if M is an elementary substructure of N; equivalently, every first-order formula with parameters from M has the same truth value in M and in N. Notation: M ≺ N or N ⪰ M.

 

 

 

 

 





 

 



 ##  [Elementary Extension](https://algebra.quantumdictionary.io/elementary-extension-1) 

  

 [![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 structure B is an elementary extension of a structure A (same first-order language) if A is an elementary substructure of B; equivalently, every first-order formula with parameters from A holds in A exactly when it holds in B. Denoted B ≻ A.