Homomorphism of universal algebras

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 $$B$$.

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 $$\phi: A$$ &rarr; $$B$$ 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 $$\phi: A $$ &rarr; $$B$$ of monoids is termed a homomorphism of monoids of $$\phi( g * h) = \phi(g) * \phi(h)$$ and \phi(e) = e.

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
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.