Factorizable Automorf’hisms in Solvable Conjoint Structures I

ing certain features of physical measurement, we have examined those conjoint structures in which there are non-trivial factorizable automorphisms. One important result is the fact that (0, q> is a factorizable automorphism iff Q = n’&r-’ where n and rr’ are solutions of the structure mapping the first component onto the second and 8 is an isomorphism between induced structures (Theorem 8). It turned out that the conjoint structure is additive iff translations of the identity of the induced concatenation structure form factorizable automorphisms (Theorem 9). And this in turn was shown to be very closely tied in with the concept of a distributive operation on one of the components (Theorem 10). The final sections combine together two additional features of physical measurement, namely, that the structures are Dedekind complete, and so have representations onto real intervals, and that the automorphisms exhibit what we have called component homogeneity and 260 R.D. Lute. M. Cohen uniqueness properties. In case the structure has an intrinsic zero and satisfies component l-point homogeneity and l-point uniqueness on its positive part, then it has a representation that is expressed in terms of unit structures (Theorem 13). The remaining theorems (15, 17, 18, and 19) combine in several ways assumptions about the smoothness of F with properties of the factorizable automorphisms that force F to be additive. In the course of this, we establish a condition for additivity of F which is related to but entails less smoothness on F than the classical result of Scheffe [lo] (Theorem 16). All of these results rest on invoking some degree of component Ior 2-point homogeneity, which has the major effect of allowing one to mix various automorphisms on the components in order to get the same transformation on the structure. There are, however, interesting structures for which the factorizable automorphisms satisfy lor 2-point homogeneity, but not component Ior 2-point homogeneity. For example, FIX, Y) = ax+(l -cr)y, x>y, O<crcl, Px+(l -Lou, XlY, O<Pc 1, has a factorizable automorphism of the form x+rx+s, y-+ry +s, F-rF+s in which exactly the same affine transformation is applied to each component. This suggests working out the theory when only homogeneity, not component homogeneity is assumed. This has been done. As it is intricate, lengthy, and the work of just one of us (Cohen) it will be published separately.