Let ,
and
be structures for a common language , and let . Then is a homomorphism from to provided that it satisfies the following:

1. For each constant , .

2. For each predicate symbol , if the arity of is , then

3. For each function symbol (or operation) , if the arity of is , then for any ,

For example, let and be (directed) graphs (the set is the set of vertices of , and is the set of vertices of , while is the relational representation of the edges of the graph
,
etc.). A homomorphism from to is a function such that for any vertices and of , and are connected by a directed edge (from to if and only if the vertices and are connected by a directed edge from to .

Another example is available in the theory of ordered groups. Let and be ordered groups. (We are using
the symbol
to denote the multiplicative inversion operation. We will drop the superscripts
and ,
and for any
(or ),
we denote
by .)
Formal application of our definition of a homomorphism in this setting indicates
that
is a homomorphism if and only if it satisfies the following:

1. .

2. For ,
.

3. For any ,
.

4. For any ,
if and only if .

(Of course, these conditions can be shown to be redundant. Hence many texts define homomorphisms with requiring the preservation of the group identity (), and with postulating the preservation of multiplicative
inversion.)

The homomorphisms of universal algebra are special cases of structure homomorphisms, and the notion of a structure homomorphism also
extends the corresponding morphism notions in categories of ordered sets and various
relational/algebraic structures.

Bell, J. L. and Slomson, A. B. Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland,
1971.Enderton, H. B. A
Mathematical Introduction to Logic. New York: Academic Press, 1972.Insall,
E. "Nonstandard Methods and Finiteness Conditions in Algebra." Ph.D. dissertation.
Houston, Texas: University of Houston, 1989.