Difference between revisions of "Homomorphism of universal algebras"
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Latest revision as of 23:43, 7 May 2008
This article defines a term related to universal algebra
Contents
Definition
Let and be two algebras in a variety of algebras. Then, a map from to is termed a homomorphism of universal algebras if where is a member of the operator domain corresponding to the variety.
The on the left is in and the on the right is in .
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 → of magmas, is a homomorphism if and only if, for every in :
The on the left is in and the on the right is in .
Here, plays the role of . Note that we have used infix notation for as opposed to prefix notation for , 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 , such that:
 viz is associative
 viz is a neutral element for
A map → of monoids is termed a homomorphism of monoids of 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 magmatheoretic homomorphisms of monoids. However, it is not true that any magmatheoretic 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 magmatheoretic homomorphism between groups is also a homomorphism of groups. Hence, we can also define a homomorphism of groups as a settheoretic map between groups that preserves the binary operation.