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

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.