Homomorphism

A homomorphism is a function between mathematical structures that preserves their specified operations or relations. In abstract algebra, it connects structures of the same type, such as groups, rings, or modules, while respecting their algebraic rules. It need not be one-to-one or onto: distinct elements may have the same image, and some elements of the target may not be reached. The defining requirement is compatibility with the structure, rather than preservation of every distinction between elements. (math.hawaii.edu)

General definition

In universal algebra, let $A$ and $B$ have the same signature: the same operation symbols, each with a specified number of arguments. A function $h:A\to B$ is a homomorphism if, for every basic $n$-ary operation $\omega$,

Thus, applying an operation and then mapping gives the same result as mapping the arguments first and then applying the corresponding operation. Constants are zero-argument operations, so designated constants must also be preserved. For example, preserving a specified constant $c$ means $h(c_A)=c_B$. (math.hawaii.edu)

Preservation of the basic operations implies preservation of every expression built from those operations. In particular, homomorphisms commute with term operations, not merely with the operations explicitly listed in the signature. (agda-algebras.universalalgebra.org)

Principal algebraic cases

Groups. In group theory, a homomorphism $h:G\to H$ satisfies

This condition implies $h(e_G)=e_H$ and $h(x^{-1})=h(x)^{-1}$. A central example is the reduction map from the additive group of integers to integers modulo $n$,

It preserves addition while identifying integers that differ by a multiple of $n$, connecting homomorphisms with modular arithmetic. Another example is the determinant map from invertible matrices over a field to the multiplicative group of nonzero field elements. (jmilne.org)

Rings. A homomorphism between rings preserves addition and multiplication:

Under the convention that rings and their homomorphisms are unital, it must also satisfy $h(1_R)=1_S$. Some treatments allow maps that do not preserve the multiplicative identity, so the convention matters. For a commutative ring $R$ and $a\in R$, evaluation of a polynomial gives a ring homomorphism

For example, evaluation at zero sends a polynomial to its constant term. (math.ucla.edu)

Vector spaces and modules. A homomorphism of vector spaces over a fixed field $F$ is a linear map:

For modules over a fixed ring, the analogous requirement is preservation of addition and the ring’s scalar action. These are instances of the same general operation-preserving definition. (math.hawaii.edu)

Image, kernel, and quotients

The image of $h:A\to B$ is

It is closed under the target’s operations and therefore forms a subalgebra. The kernel describes what the map identifies. For a group homomorphism it is the set mapped to the identity; for a ring homomorphism it is the set mapped to zero. These form, respectively, a normal subgroup and an ideal. (math.hawaii.edu)

For a linear map, the kernel is $\{v:T(v)=0\}$, also called its null space. In general universal algebra, however, there may be no distinguished zero or identity. The kernel is then defined as a relation:

It is an equivalence relation compatible with every operation, called a congruence relation. A homomorphism is injective exactly when this relation identifies no distinct elements. (math.hawaii.edu)

The first isomorphism theorem states that quotienting by the identifications made by a homomorphism produces a structure isomorphic to its image:

For groups this becomes

with the isomorphism sending the coset $g\ker h$ to $h(g)$. Thus, a homomorphism factors into a quotient map, an isomorphism onto its image, and inclusion of that image into the target. (math.hawaii.edu)

Quotient maps also have a universal property. If a group homomorphism $f:G\to H$ sends a normal subgroup $N$ to the identity, there is a unique homomorphism $\bar f:G/N\to H$ satisfying $f=\bar f\circ\pi$, where $\pi(g)=gN$. (jmilne.org)

Composition and related terminology

The composition of homomorphisms is a homomorphism, and identity functions preserve structure. Homomorphisms also carry subalgebras to subalgebras and are determined by their values on a generating set. (math.hawaii.edu)

Several related terms specify additional properties:

• An endomorphism is a homomorphism from a structure to itself.
• An isomorphism has an inverse that also preserves structure. For purely algebraic structures of a fixed signature, every bijective homomorphism has this property.
• An automorphism is an isomorphism from a structure to itself. (math.hawaii.edu)

In category theory, a monomorphism is defined by left cancellation and an epimorphism by right cancellation. These definitions should not simply be replaced by “injective” and “surjective” in every category. For example, the inclusion $\mathbb Z\hookrightarrow\mathbb Q$ is an epimorphism of commutative unital rings but is not surjective: every rational number is determined by integers and their multiplicative inverses, so two ring maps out of $\mathbb Q$ that agree on $\mathbb Z$ agree everywhere. (stacks.math.columbia.edu)

Relational structures and graphs

The term extends to structures defined by relations rather than operations. In graph theory, a graph homomorphism maps vertices while preserving adjacency:

It need not preserve nonadjacency. A proper coloring of a loopless graph with $k$ colors is precisely a homomorphism to the complete graph $K_k$. (maths.tcd.ie)

Unlike the purely algebraic case, a bijective graph homomorphism need not be an isomorphism: its inverse may fail to preserve adjacency. For instance, a bijection from the vertices of a three-vertex path to a triangle preserves all edges of the path, but the reverse map cannot preserve all triangle edges. (maths.tcd.ie)

Scope of preservation

The structure being preserved must always be specified. Preserving addition alone does not establish a ring homomorphism; preserving vector addition alone does not establish linearity over a chosen field. Additional topological structure likewise introduces additional requirements beyond algebraic preservation. (math.ucla.edu)

A homeomorphism is a different concept: a bijective continuous map with continuous inverse, expressing equivalence of topological spaces rather than merely preservation of algebraic operations. The similarity of the two names does not make their defining conditions interchangeable. (ocw.mit.edu)

References

1. Lectures on Universal Algebramath.hawaii.edu
2. A Course in Universal Algebramath.hawaii.edu
3. Group Theoryjmilne.org
4. Operations — The Agda Universal Algebra Libraryagda-algebras.universalalgebra.org
5. Abstract Algebra — Chapter 3 Ring theorymath.ucla.edu
6. AATA Homomorphisms, Normal Subgroups and Quotientsntouikan.ext.unb.ca
7. Section 10.107: Epimorphisms of rings — The Stacks projectstacks.math.columbia.edu
8. Homomorphismsmaths.tcd.ie
9. RES.18-012 (Spring 2022) Full Lecture Notes: Algebra II Student Notesocw.mit.edu