Classification of Concatenation Measurement Structures according to Scale Type*’

A relational structure is said to be of scale type (M, N) iff M is the largest degree of homogeneity and N the least degree of uniqueness (Narens, Theory and Decision, 1981, 13, l-70; Journal qf Mathematical Psvchology, 1981, 24, 249-275) of its automorphism group. Roberts (in Proceedings qf the.first Hoboken Symposium on graph theory, New York: Wiley, 1984; in Proceedings of the fifth international conlerence on graph theory and its applications, New York: Wiley, 1984) has shown that such a structure on the reals is either ordinal or M is less than the order of at least one defining relation (Theorem 1.2). A scheme for characterizing N is outlined in Theorem 1.3. The remainder of the paper studies the scale type of concatenation structures (X, 2, ‘ ), h w ere 2 is a total ordering and 0 is a monotonic operation. Section 2 establishes that for concatenation structures with M > 0 and N < co the only scale types are (1, 1). (1, 2) and (2, 2), and the structures for the last two are always idempotent. Section 3 is concerned with such structures on the real numbers (i.e., candidates for representations), and it uses general results of Narens for real relational structures of scale type (M, M) (Theorem 3.1) and of Alper (Journal of Mathematical P.yychology, 1985, 29, 73-81) for scale type (I, 2) (Theorem 3.2). For M>O. concatenation structures are all isomorphic to numerical ones for which the operation can be written ~0 y = ,P~(.x/v), where ,f is strictly increasing and ,f(.u)/ x is strictly decreasing (unit structures). The equation ,f’(+) = f(s)” is satisfied for all I as follows: for and only for p = 1 in the (I, 1) case; for and only for p = k”, k > 0 fixed, and n ranging over the integers, in the (1. 2) case; and for all p > 0 in the (2,2) case (Theorems 3.9, 3.12, and 3.13). Section 4 examines relations between concatenation and conjoint structures, including the operation induced on one component by the ordering of a conjoint structure and the concept of an operation on one component being distributive in a conjoint structure. The results, which are mainly of interest in proving other