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STRUCTURAL PROPERTIES AND MEASUREMENT THEORY OF CERTAIN SETS ADMITTING A CONCATENATION OPERATION
Author(s) -
Kristof Walter
Publication year - 1968
Publication title -
british journal of mathematical and statistical psychology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.157
H-Index - 51
eISSN - 2044-8317
pISSN - 0007-1102
DOI - 10.1111/j.2044-8317.1968.tb00410.x
Subject(s) - concatenation (mathematics) , commutative property , uniqueness , mathematics , metric (unit) , string (physics) , discrete mathematics , algebra over a field , pure mathematics , combinatorics , mathematical analysis , operations management , economics , mathematical physics
This paper is a contribution to the theory of unidimensional scaling based on a concatenation operation defined for pairs of objects from certain continuous and ordered sets M. Inferences about structural properties of M , such as reflexivity and commutativity of the concatenation operation, are obtained by means of an entirely general non‐metric treatment. It is shown that reflexivity holds either throughout or just for a single element and that commutativity of two different elements holds either generally or in no instance. These statements are independent. The study employs concatenation operations defined in an ad hoc way which formally and necessarily satisfy the reflexivity requirement and then also that of commutativity. By this means, the existence and essential uniqueness of scales of M with linear expressions for the various concatenation operations are established. Rules are derived for the way in which the parameters entering into these expressions can change with admissible scale transformations.