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Conditions Equivalent to Unit Representations of Ordered Relational Structures.

R. Duncan Luce¹

¹Institute for Mathematical Behavioral Sciences, University of California, Irvine

Journal of Mathematical Psychology

|February 17, 2001

Summary

This study explores automorphism subgroups in ordered structures, finding conditions for numerical representations. These conditions, including homogeneous and Archimedean properties, are shown to be sufficient for specific scale types.

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Area of Science:

Order theory
Group theory
Mathematical structures

Background:

Investigates subgroups of automorphisms in linearly ordered relational structures.
Connects properties of automorphism groups to numerical representations of structures.

Purpose of the Study:

To identify conditions equivalent to translations forming a homogeneous, Archimedean ordered group.
To establish sufficiency of necessary conditions for numerical representation.

Main Methods:

Analysis of automorphism subgroups and their induced asymptotic order.
Examination of homogeneity, Archimedean properties, and convex subgroups.

Main Results:

Theorem 5: Connected asymptotic order, homogeneity, Archimedean translations, and relative Archimedean dilations are sufficient for numerical representation.
Identifies the condition for dilations being Archimedean relative to all automorphisms.

Conclusions:

The established conditions are sufficient for numerical representations between interval and ratio scales.
Results do not simplify proofs for Dedekind complete cases; parallel automorphisms are also examined.