A general representation for internal proportional cornbinatorial measurement systems when the operation is not necessari!Y closed

The aim of this paper is to give one kind of internal proportional systems with general representation and without closure and finiteness assumptions. First, we introduce the notions of internal proportional system and of general representation. Second, we briefly review the existing results which motivate our generalization. Third, we present the new systems, characterized by the fact that the linear order induced by the comparison weak order ≥ at the level of equivalence classes is also a weIl order. We prove the corresponding representation theorem and make some comments on strong limitations of uniqueness; we present in an informal way a positive result, restricted uniqueness for what we call connected objects. We conclude with some final remarks on the property that characterizes these systems and on three possible empirical applications