##  [Free Algebra](/index.php/free-algebra) 

  ##  [Free Algebra](https://puremath.quantumdictionary.io/free-algebra-0) 

  

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

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebra in a given signature and variety generated by a set of free generators with no defining relations other than the identities required by the signature and the variety; equivalently, an object F(X) equipped with an inclusion of the generator set X such that every function from X into any algebra A of the same variety extends uniquely to a homomorphism F(X) → A.

 

 

 

 

 





 

 



 ##  [Free Algebra](https://algebra.quantumdictionary.io/free-algebra-1) 

  

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

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebra in a variety generated freely by a set of generators; it satisfies the universal mapping property that every function from the generator set to any algebra of the variety extends uniquely to a homomorphism from the free algebra.