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.