AN ALTERNATIVE MEASUREMENT STRUCTURE: THE POSITIVELY ORDERED PARTIAL SEMIGROUP WITH IDENTITY (HALFGROUPOID)
Abstract
A set of axioms which define a positively ordered partial semigroup with identity, an alternative measurement system, is introduced. The positively ordered partial semigroup with identity is defined by a set A of objects, an order among those objects, an operation of concatenation (a binary operation from some subset of A x A into A), and axioms which delineate the properties of the order among the objects and the concatenation operation. One common feature of axiomatic systems of measurement (with the exception of the abelian positive ordered quasi-group) is the tendency to place stringent requirements on the domain of the concatenation operation. These requirements seem to limit the systems' potential applications; in particular, they are recognized to limit their economic applications. The impetus behind the formulation of the axioms which define the positively ordered partial semigroup with identity was to create a system with limited restrictions on the domain of the concatenation operation. Comparisons and contrasts are made between the axioms of this system and axioms which define other measurement systems. In particular, similarities and differences between the axioms which define a positively ordered partial semigroup with identity and those which define Krantz, Luce, Suppes, and Tversky's Archimedean ordered semigroup and Moore's abelian positive ordered quasi-group are discussed. Through an indirect method of proof, it is shown that a positively ordered partial semigroup with identity has an image in the nonnegative additive real numbers. Specifically, this is accomplished by showing that such a structure can be embedded in a fully ordered semigroup with certain characteristics which is known to be homomorphically embeddable in the nonnegative additive real numbers. Finally, the relationship between the positively ordered partial semigroup with identity and the Archimedean ordered semigroup is discussed. In particular, it is shown that a structure which resembles an Archimedean ordered semigroup is a positively ordered partial semigroup with identity.
Degree
Ph.D.
Subject Area
Economic theory
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