##  [Definable Closure](/definable-closure) 

  ##  [Definable Closure](https://mathlogic.quantumdictionary.io/definable-closure-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

Given a structure M and a subset A, the definable closure dcl(A) is the set of all elements of M that are uniquely specified (in M) by some first‑order formula with parameters from A; equivalently, elements fixed by every automorphism of M that fixes A pointwise.

 

 

 

 

 





 

 



 ##  [Definable Closure](https://algebra.quantumdictionary.io/definable-closure-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

The definable closure dcl(A) of a set A in a structure M is the set of elements of M that are uniquely specified by some first-order formula with parameters from A; equivalently, b ∈ dcl(A) iff there is a formula φ(x,a) with a from A such that M ⊨ φ(b,a) and M ⊨ ∀x(φ(x,a) → x=b).