Homomorphism
[1] The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).
The notation for the operations does not need to be the same in the source and the target of a homomorphism.
It is even an isomorphism (see below), as its inverse function, the natural logarithm, satisfies
where r is a real number, then f is a homomorphism of rings, since f preserves both addition:
Due to the different names of corresponding operations, the structure preservation properties satisfied by
Several kinds of homomorphisms have a specific name, which is also defined for general morphisms.
An isomorphism between algebraic structures of the same type is commonly defined as a bijective homomorphism.
[3]: 135 The endomorphisms of an algebraic structure, or of an object of a category, form a monoid under composition.
The endomorphisms of a vector space or of a module form a ring.
For algebraic structures, monomorphisms are commonly defined as injective homomorphisms.
[3]: 134 [4]: 29 In the more general context of category theory, a monomorphism is defined as a morphism that is left cancelable.
These two definitions of monomorphism are equivalent for all common algebraic structures.
More precisely, they are equivalent for fields, for which every homomorphism is a monomorphism, and for varieties of universal algebra, that is algebraic structures for which operations and axioms (identities) are defined without any restriction (the fields do not form a variety, as the multiplicative inverse is defined either as a unary operation or as a property of the multiplication, which are, in both cases, defined only for nonzero elements).
In particular, the two definitions of a monomorphism are equivalent for sets, magmas, semigroups, monoids, groups, rings, fields, vector spaces and modules.
For example, an injective continuous map is a monomorphism in the category of topological spaces.
For proving that, conversely, a left cancelable homomorphism is injective, it is useful to consider a free object on
Two such formulas are said equivalent if one may pass from one to the other by applying the axioms (identities of the structure).
This defines an equivalence relation, if the identities are not subject to conditions, that is if one works with a variety.
[3]: 134 [4]: 43 On the other hand, in category theory, epimorphisms are defined as right cancelable morphisms.
A surjective homomorphism is always right cancelable, but the converse is not always true for algebraic structures.
Algebraic structures for which there exist non-surjective epimorphisms include semigroups and rings.
The most basic example is the inclusion of integers into rational numbers, which is a homomorphism of rings and of multiplicative semigroups.
[5][7] A wide generalization of this example is the localization of a ring by a multiplicative set.
the last implication is an equivalence for sets, vector spaces, modules, abelian groups, and groups; the first implication is an equivalence for sets and vector spaces.
In the case of vector spaces, abelian groups and modules, the proof relies on the existence of cokernels and on the fact that the zero maps are homomorphisms: let
of the identity element of this operation suffices to characterize the equivalence relation.
This structure type of the kernels is the same as the considered structure, in the case of abelian groups, vector spaces and modules, but is different and has received a specific name in other cases, such as normal subgroup for kernels of group homomorphisms and ideals for kernels of ring homomorphisms (in the case of non-commutative rings, the kernels are the two-sided ideals).
Let L be a signature consisting of function and relation symbols, and A, B be two L-structures.
[8] Homomorphisms are also used in the study of formal languages[9] and are often briefly referred to as morphisms.
Here the monoid operation is concatenation and the identity element is the empty word.
