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Homomorphism of universal algebras

This article defines a term related to universal algebra

Contents

Definition

Let A and B be two algebras in a variety of algebras. Then, a map \phi from A to B is termed a homomorphism of universal algebras if \phi(f(a_1,a_2,...a_n)) = f(\phi(a_1),\phi(a_2),...,\phi(a_n)) where f is a member of the operator domain corresponding to the variety.

The f on the left is in A and the f on the right is in B.

Examples

Homomorphism of magmas

Consider the variety of magmas: a magma is a set equipped with a binary operation. The operator domain here consists of a single operator: the binary operation of multiplication (denoted as *). Thus, given a map \phi: AB of magmas, \phi is a homomorphism if and only if, for every g,h in A:

\phi(g * h) = \phi(g) * \phi(h)

The * on the left is in A and the * on the right is in B.

Here, * plays the role of f. Note that we have used infix notation for * as opposed to prefix notation for f, which is why the expression looks somewhat different.

Homomorphism of monoids

Consider the variety of monoids: a monoid is a set equipped with a binary operation *, as well as a constant called the neutral element e, such that:

A map \phi: A B of monoids is termed a homomorphism of monoids of \phi( g * h) = \phi(g) * \phi(h) and \phi(e) = e</math>.

Note that since every monoid is also a magma (by only looking at the binary operation) we can also talk of magma-theoretic homomorphisms of monoids. However, it is not true that any magma-theoretic homomorphism is also a homomorphism of monoids. In particular, the neutral element may not go to the neutral element.

Homomorphism of groups

Further information: homomorphism of groups

A homomorphism of groups is a map from one group to another that preserves: the binary operation, the inverse operation and the neutral element. It turns out that any magma-theoretic homomorphism between groups is also a homomorphism of groups. Hence, we can also define a homomorphism of groups as a set-theoretic map between groups that preserves the binary operation.