What does endomorphism mean?
An endomorphism is a mathematical map that sends a structure to itself while preserving the relevant structure. For example, a group endomorphism takes elements of a group and combines them in a way that respects the group operation, while a linear endomorphism of a vector space can be represented by a matrix. In category theory, the term is more general: an endomorphism is any morphism whose source and target are the same object. The important distinction is that an endomorphism does not have to be reversible or one-to-one. If it is bijective, it is called an automorphism. The word is specialized but useful because it names a common mathematical idea and helps distinguish self-maps from maps between different structures.
nounIn mathematics, an endomorphism is a morphism or homomorphism from an algebraic structure, or a category's object, to itself. It need not be invertible; an invertible endomorphism is called an automorphism.
- A homomorphism from an algebraic structure to itself, such as a group, ring, or vector space.
- In category theory, a morphism whose source and target are the same object.
"The identity map is an endomorphism of every vector space."
"The identity map is an endomorphism of every vector space."
"A linear endomorphism of a finite-dimensional vector space can be represented by a matrix."
The plural is regular and is formed by adding -s.
"The group has several distinct endomorphisms."
An endomorphism is a mathematical self-portrait: a structure acting on itself without needing to change into something else.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Where does endomorphism come from?
Endomorphism was formed in mathematical terminology by combining endo-, meaning “within” or “inside,” with morphism, a term used for a structure-preserving mapping. Its formation parallels words such as automorphism, isomorphism, and polymorphism, which share the morphism element. The prefix endo- comes from Greek endon, “within,” while the broader mathematical family reflects the Greek root morphē, “form.”
Related word forms
How endomorphism is actually used
Highly specialized mathematical terminology, used primarily in algebra, category theory, and related areas. It is normally encountered in formal exposition rather than general conversation.
Easily confused with endomorphism
An endomorphism is a structure-preserving map from an object to itself, whereas an automorphism is an invertible endomorphism.
An endomorphism maps a structure to itself, while an isomorphism is a structure-preserving bijection between two structures that may be different.