Strong Homomorphisms and Embeddings

Homomorphisms are usually defined as structure preserving mappings from one model to another. Representation theorems are taken to establish the existence of a homomorphism between a qualitative first-order structure endowed with some empirical relations and some sort of numerical first order structure. The classic example is in the measurement of hedonic utility using introspection. In that case, we describe the axiomatic conditions ensuring the existence of a mapping from a structure to the structure of reals with their standard ordering . Here is meant to be a (usually finite) set of alternatives or choices and the relation is meant to encode the introspectively accessible relation that something feels better than something else.

I have been thinking that this notion of a homomorphism was exactly the same as in model theory but it turns out there are some subtleties. In model theory, usually we say that if we have two structures and in the same signature (which is a set of constant symbols, relation symbols, and function symbols), then a homomorphism from , the domain of , to , the domain of is a function satisfying the following conditions:

  • (i) For any constant symbol in , is
  • (ii) For any -ary function symbol in and ,

  • (iii:a) For any -ary relation symbol in and ,

The superscripts here indicate how the symbols are interpreted in the respective structures with objects or -tuples of objects. The conditions taken together are less demanding conditions than what is usually meant, it appears, in the theory of measurement. Here, we replace the third condition with

  • (iii:b) For any -ary relation symbol in and ,

In model theory, a mapping satisfying (i), (ii), and (iii:b) is called a strong homomorphism. The condition ensures that if two objects, for example, are not -related in then the will remain unrelated by in the mapping to .

If we add the condition that , a strong homomorphism, is an injection then the map is usually called an embedding. Likewise if a strong homomorphism is a bijection then it is an isomorphism.