Structure Homomorphism
In logic, the term "homomorphism" is used in a manner similar to but a bit different from its usage in abstract algebra. The usage in logic is a special case of a "morphism" from category theory.
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
![P^B=<span style=]() {h(a_1),...,h(a_n)|(a_1,...,a_n) in P^A}. " src="https://mathworld.wolfram.com/images/equations/StructureHomomorphism/NumberedEquation1.svg" style="height:23px; width:295px" /> |
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.
REFERENCES
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.
