Ordering orders

Abstract

This paper is concerned with structures of the form (A, 0) where A is a nonempty set and O a set of order relations on A. In particular we investigate their representability as a product structure with a weak order defined on a Cartesian product satisfying independence in the Conjoint Measurement sense. It is shown by a number of examples that systems of this kind, i.e. (A, B)-structures are often encountered. Conditions are formulated under which the set O can be partially ordered, which is a necessary requirement for its representability as a conjoint structure. The investigation of the relation between the automorphism group of (A, 0) and the automorphism group of the representing product structure gives rise to the introduction of a new class of automorphisms, generalizing the concept of a factorizable automorphism invented by Luce & Cohen. The generalized factorizable automorphisms are proven to form a group which contains the ordinary factorizable automorphisms as a normal subgroup. (C) 1998 Elsevier Science B.V. All rights reserved.