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# 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.

Homomorphism

This entry contributed by Matt Insall (author's link)

## 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.

## Referenced on Wolfram|Alpha

Structure Homomorphism

## Cite this as:

Insall, Matt. "Structure Homomorphism." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/StructureHomomorphism.html