Homomorphism
![Monoid homomorphism f from the monoid (N, +, 0) to the monoid (N, ×, 1), defined by f(x) = 2x. It is injective, but not surjective.](/Images/godic/202501/18/Exponentiation_as_monoid_homomorphism_svg.svg1223.png")
![Relationships between different kinds of homomorphisms. Hom = set of Homomorphisms, Mon = set of Monomorphisms, Epi = set of Epimorphisms, Iso = set of Isomorphisms, End = set of Endomorphism, Aut = set of Automorphisms. Notice that: Mon ∩ Epi = Iso, Iso ∩ End = Aut. The sets (Mon ∩ End) \ Aut and (Epi ∩ End) \ Aut contain only homomorphisms from some infinite structures to themselves.](/Images/godic/202501/18/Morphisms3.svg1223.png")
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). The word homomorphism comes from the ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape". Isomorphisms, automorphisms, and endomorphisms are special types of homomorphisms.