Homomorphism

related topics
{math, number, function}
{language, word, form}

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 Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "shape".

Contents

Definition and illustration

Definition

The definition of homomorphism depends on the type of algebraic structure under consideration. Particular definitions of homomorphism include the following:

The common theme is that a homomorphism is a function between two algebraic objects that respects the algebraic structure.

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If G and H are groups, a homomorphism from G to H is a function ƒG → H such that f(g_1 * g_2) = f(g_1) *' f(g_2)\,\! for any elements g1g2 ∈ G, where * denotes the operation in G and *' denotes the operation in H.

Full article ▸

related documents
Infimum
Division ring
Principal ideal domain
Semi-continuity
Cofinality
Ternary numeral system
Iterative method
Elliptic function
Upper and lower bounds
Conjugacy class
Counting sort
Lambert W function
ZPP
Cyclone (programming language)
Shannon–Fano coding
Multiplication table
Möbius function
Pointless topology
Commutator
Banach algebra
Dedekind cut
Five lemma
Elementary function
Square-free integer
Pseudocode
Distributivity
Caesar cipher
Intermediate value theorem
HMAC
Arithmetic function