Homomorphism of universal algebras

This article defines a term related to universal algebra

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 $\phi :A$ → $B$ 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:

• $a*(b*c)=(a*b)*c$ viz $*$ is associative
• $a*e=e*a=a$ viz $e$ is a neutral element for $*$

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.