Measurement structures with archimedean ordered translation groups

Abstract

The paper focuses on three problems of generalizing properties of concatenation structures (ordered structures with a monotonic operation) to ordered structures lacking any operation. (1) What is the natural generalization of the idea of Archimedeaness, of commensurability between large and small? (2) What is the natural generalization of the concept of a unit concatenation structure in which the translations (automorphisms with no fixed point) can be represented by multiplication by a constant? (3) What is the natural generalization of a ratio scale concatenation structure being distributive in a conjoint one, which has been shown to force a multiplicative representation of the latter and the product-of-powers representation of units found in physics? It is established (Theorems 5.1 and 5.2) that for homogeneous structures, the latter two questions are equivalent to it having the property that the set of all translations forms a homogeneous Archimedean ordered group. A sufficient condition for Archimedeaness of the translations is that they form a group, which is equivalent to their being 1-point unique, and the structure be Dedekind complete and order dense (Theorems 2.1 and 2.2). It is suggested that Archimedean order of the translations is, indeed, also the answer to the first question. As a lead into that conclusion, a number of results are reported in Section 3 on Archimedeaness in concatenation structures, including for positive structures sufficient conditions for several different notions of Archimedeaness to be equivalent. The results about idempotent structures are fragmentary.