Tight and loose value automorphisms (Q757409)

In their work, \textit{T.M. Alper}, \textit{R. D. Luce} and \textit{L. Narens} [J. Math. Psychol. 24, 249-275 (1981; Zbl 0496.92020), 27, 44-125 (1983; Zbl 0559.90025), 29, 1-72, 73-81 (1985; Zbl 0578.92028, Zbl 0602.92022), 30, 391-415 (1986; Zbl 0603.06010), 31, 135-154 (1987; Zbl 0626.06018), Order 4, 165-189 (1987; Zbl 0629.08001), Theory Decis. 13, 1-70 (1981; Zbl 0497.00021)] have developed a classification of relational systems and of the corresponding measurement problems by using two concepts defined in terms of the group of automorphisms: degree of homogeneity, h, and degree of uniqueness, u. They study these parameters h and u for special relational systems, with the goal of determining what values of h, u and (h,u) can be attained. In this paper, we introduce the investigation of the pairs (h,u) for certain kinds of digraphs and we introduce variants on the parameters h and u. By assuming much less structure than is assumed by Alper, Luce, and Narens, and by introducing a finiteness assumption, we shall obtain a classification of objects with many more types. We show that for the Alper-Luce-Narens parameters and several variants, if \(h>1\) and \(u<\infty\), then all pairs (h,u) with \(h\leq u\) can be attained. For several other variants of these parameters, if \(h>1\) and \(u<\infty\), then all pairs (h,u) with \(h=u\) or \(h=u-1\) can be attained.