*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

*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

*homomorphism*from

**(i)**For any constant symbolc in\tau ,\rho(c^{\mathbf{A}}) isc^{\mathbf{B}} **(ii)**For anyn -ary function symbolf in\tau andx_1, . . ., x_n \in \mathrm{A} ,\rho(f^{\mathbf{A}}(x_1, . . ., x_n)) = f^{\mathbf{B}}(\rho(x_1), . . ., \rho(x_n)) **(iii:a)**For anyn -ary relation symbolR in\tau andx_1, . . ., x_n \in \mathrm{A} ,R^{\mathbf{A}}(x_1, . . ., x_n) \Longrightarrow R^{\mathbf{B}}(\rho(x_1), . . ., \rho(x_n))

The superscripts here indicate how the symbols are interpreted in the respective structures with objects or

**(iii:b)**For anyn -ary relation symbolR in\tau andx_1, ..., x_n \in \mathrm{A} ,R^{\mathbf{A}}(x_1, ..., x_n) \Longleftrightarrow R^{\mathbf{B}}(\rho(x_1), ..., \rho(x_n))

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

If we add the condition that

*embedding*. Likewise if a strong homomorphism is a bijection then it is an isomorphism.