# Conditions Equivalent to Unit Representations of Ordered Relational Structures.

@article{Luce2001ConditionsET, title={Conditions Equivalent to Unit Representations of Ordered Relational Structures.}, author={R. Duncan Luce}, journal={Journal of mathematical psychology}, year={2001}, volume={45 1}, pages={ 81-98 } }

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… Expand

#### Topics from this paper

#### 9 Citations

Scale Type (N, N) and an Order-Based Topology Induced on the Automorphism Group.

- Mathematics, Medicine
- Journal of mathematical psychology
- 2000

The automorphism group is shown to be a topological group that acts topologically on the base set A of the structure and on its rank-ordered Cartesian products and the property of local compactness of the automorphic group is derived and related results proved. Expand

Generalized factorizable automorphisms in n-component conjoint structures

- Mathematics
- 2003

Abstract The group of generalized factorizable automorphisms (Math. Soc. Sci. 36 (1998) 91) of a n-component conjoint structure, which satisfies certain solvability assumptions and the Thomsen… Expand

On an algebraic definition of laws

- Mathematics
- 2014

Abstract An algebraic definition of laws is formulated, motivated by analyzing points in Euclidean geometry and from considerations of two physical examples, Ohm’s law and Newton’s second law. Simple… Expand

Beyond the metrological viewpoint.

- Medicine
- Studies in history and philosophy of science
- 2020

This paper presents various cases where RTM can be said to investigate measurement without providing any measurement procedure, and illustrates some of the questions which the philosophy of measurement can explore, when it is ready to go beyond the metrological viewpoint. Expand

Objective Physiological Measurements but Not Subjective Reports Moderate the Effect of Hunger on Choice Behavior

- Psychology, Medicine
- Front. Psychol.
- 2018

It is found that hunger increases risk-seeking behavior in a lottery-choice task, modifies levels of vindictiveness in a social decision-making task, but does not have a detectible effect on economic inconsistency in a budget-set choice task. Expand

Manipulating measurement scales in medical statistical analysis and data mining: A review of methodologies

- Medicine
- Journal of research in medical sciences : the official journal of Isfahan University of Medical Sciences
- 2014

This work has presented two ordinal–variables clustering examples, as more challenging variable in analysis, using Wisconsin Breast Cancer Data, and found that by using appropriate clustering algorithm based on the measurement scale of the variables in the study, high performance is granted. Expand

Measurement, Representational Theory of

- Mathematics
- 2001

This article is reproduced from the previous edition, volume 14, pp. 9442–9448, © 2001, Elsevier Ltd.

Scales of Measurement and Choice of Statistical Methods

- Mathematics, Computer Science
- International Encyclopedia of Statistical Science
- 2011

#### References

SHOWING 1-10 OF 21 REFERENCES

Uniqueness and homogeneity of ordered relational structures

- Mathematics
- 1986

Abstract There are four major results in the paper. (1) In a general ordered relational structure that is order dense, Dedekind complete, and whose dilations (automorphisms with fixed points) are… Expand

Singular points in generalized concatenation structures that otherwise are homogeneous

- Mathematics
- 1992

Abstract A generalized concatenation structure X is any ordered relational structure in which at least one relation is a function of two or more variables that is monotonically increasing in each of… Expand

Measurement structures with archimedean ordered translation groups

- Mathematics
- 1987

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… Expand

A classification of all order-preserving homeomorphism groups of the reals that satisfy finite uniqueness

- Mathematics
- 1987

The only real relational structures of scale type (m, n) with 1 ≤ m ≤ n < ∞ are of scale type (1, 1), (1, 2), and (2, 2), and so are conjugate to structures whose automorphism groups are subgroups of… Expand

Groups of order-automorphisms of the rationals with prescribed scale type

- Mathematics
- 1989

Abstract It is shown that, for any given m and n with m n (where n = ∞ is allowed), there is a group of order-preserving permutations of the rational numbers whose degrees of homogeneity and… Expand

Classification of concatenation measurement structures according to scale type

- Mathematics
- 1985

Abstract A relational structure is said to be of scale type ( M , N ) iff M is the largest degree of homogeneity and N the least degree of uniqueness ( Narens, Theory and Decision , 1981, 13 , 1–70;… Expand

Sharply multiply homogeneous permutation groups, and rational scale types

- Mathematics
- 1996

It is shown that for every positive integer k, the free group of countably infinite rank has a faithful permutation representation s a sharply fc-homogeneous group of order-automorphisms of the… Expand

Fundamental unit structures: A theory of ratio scalability☆

- Mathematics
- 1979

A theory of nonassociative measurement structures is developed which produces a natural generalization of associative measurement (i.e., extensive structures), and representation and uniqueness… Expand

On the scales of measurement

- Mathematics
- 1981

Abstract Let X = 〈 X , ≧, R 1 , R 2 …〉 be a relational structure, 〈 X, ≧ 〉 be a Dedekind complete, totally ordered set, and n be a nonnegative integer. X is said to satisfy n -point homogeneity if… Expand

A note on real measurement structures of scale type (m, m + 1)

- Mathematics
- 1985

Abstract Real relational structures of scale type (1, 2) can be represented in the reals in such a way that the induced automorphism group is properly contained in the affine group and itself… Expand