##  [Term Algebra](/term-algebra) 

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

  

 [![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 & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The free algebra of syntactic terms built from a given signature and set of variables: elements are formal term trees and operations are term-forming function symbols; it satisfies the universal property of free objects in the category of algebras for that signature.

 

 

 

 

 





 

 



 ##  [Term Algebra](https://algebra.quantumdictionary.io/term-algebra-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 free algebra over a signature generated by a set of variables whose elements are syntactic terms built from operation symbols and variables, taken without imposing any identities beyond the syntactic formation rules.