Conditions Equivalent to Unit Representations of Ordered Relational Structures.

This paper studies several concepts about subgroups of automorphisms of linearly ordered relational structures. In particular, it focuses on conditions that are equivalent to the translations (automorphisms with no fixed points plus the identity) forming a homogeneous, Archimedean ordered group under the asymptotic order. For the automorphisms of an ordered relational structure, these properties are equivalent to the structure having a numerical representation whose scale type lies between, but does not exclude, the ratio and interval types (Alper, 1987; Luce, 1987; Narens, 1981a, 1981b). One result of this paper (Theorem 5) is that for the automorphisms of an ordered relational structure the following necessary conditions for such a representation are also sufficient: the asymptotic order induced on the automorphisms is connected, the structure is homogeneous, the translations are Archimedean, and the dilations (i.e., automorphisms with fixed points) are Archimedean relative to all automorphisms. The last property is equivalent to there being at most one proper, nontrivial convex subgroup. Contrary to hope, these results do not seem to lead to a simpler proof than Alper's for the Dedekind complete case. The paper concludes with an examination of the structure of parallel automorphisms. Copyright 2001 Academic Press.