General framework of structural similarity between system models

The structural similarity between system models has been investigated so far solely based on the concept of the homomorphism which is defined only between system models of ‘the same type’. However, this is not satisfactory for the development of systems theory. This paper extends the concept of the homomorphism to investigate the structural similarity between system models, not only of the same type but of different types, by introducing F-morphisms, and examines what kind of system properties are preserved in terms of the extended homomorphisms. This paper deals with three kinds of properties to be preserved: generators, a set of axioms Σ, and all sentences satisfied in the system model Th(M); it also provides six morphisms: homomorphisms, Σ-homomorphisms, S-homomorphisms, F-morphisms, ΣF-morphisms and SF-morphisms, for the three cases with respect to system models of the same type and of different types. Convertial homoniorphisms are regarded as special cases of an F-morphism. Finally an F-morphism theorem for system models of different types is proven, which corresponds to the homomorphism theorem for system models of the same type.