##  [Elementary Substructure](/elementary-substructure) 

  ##  [Elementary Substructure](https://mathlogic.quantumdictionary.io/elementary-substructure-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 substructure M of a structure N (in the same signature) that preserves the truth of every first-order formula with parameters from M; formally, for every first-order formula φ(x1,..,xn) and every tuple a from M, N ⊨ φ(a) if and only if M ⊨ φ(a). Often denoted M ≺ N.

 

 

 

 

 





 

 



 ##  [Elementary Substructure](https://algebra.quantumdictionary.io/elementary-substructure-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 substructure A of a structure B (in the same first-order language) such that for every first-order formula φ(x1,...,xn) and every tuple a from A, B ⊨ φ(a) if and only if A ⊨ φ(a). Commonly denoted A ≺ B.