In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra.

The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory.

A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.

A homomorphism is a map between two algebraic structures of the same type (e.g. two groups, two fields, two vector spaces), that preserves the operations of the structures. This means a map ${\displaystyle f:A\to B}$ between two sets ${\displaystyle A}$, ${\displaystyle B}$ equipped with the same structure such that, if ${\displaystyle \cdot }$ is an operation of the structure (supposed here, for simplification, to be a binary operation), then

$${\displaystyle f(x\cdot y)=f(x)\cdot f(y)}$$

for every pair ${\displaystyle x}$, ${\displaystyle y}$ of elements of ${\displaystyle A}$. One says often that ${\displaystyle f}$ preserves the operation or is compatible with the operation.

Formally, a map ${\displaystyle f:A\to B}$ preserves an operation ${\displaystyle \mu }$ of arity ${\displaystyle k}$, defined on both ${\displaystyle A}$ and ${\displaystyle B}$ if