##  [Homomorphism](/homomorphism) 

  ##  [Homomorphism](https://natural.quantumdictionary.io/homomorphism-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A map between algebraic structures that preserves the operations and relations specified by those structures (e.g., f(ab)=f(a)f(b) for group homomorphisms).

 

 

 

 

 





 

 



 ##  [Homomorphism](https://puremath.quantumdictionary.io/homomorphism-1) 

  

 [![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

A homomorphism is a map between algebraic structures of the same signature that preserves the defining operations and relations (e.g., group homomorphism preserves multiplication, linear map preserves addition and scalar multiplication).

 

 

 

 

 





 

 



 ##  [Homomorphism](https://algebra.quantumdictionary.io/homomorphism-2) 

  

 [![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-preserving map between two algebraic objects (for example groups, rings, modules, or vector spaces) that commutes with the defining operations: for an operation •, f(x •_A y) = f(x) •_B f(y).