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

Area of Science:

• Measurement theory
• Group theory
• Topology

Background:

• Measurement structures are defined by ordered sets with relations.
• Scale types are determined by the homogeneity (M) and uniqueness (N) of their automorphism group.
• This paper focuses on the case where 1 ≤ M = N < ∞.

Purpose of the Study:

• To investigate the topological properties of automorphism groups for measurement structures.
• To generalize the Alper-Narens theorem for scale types.
• To interpret measurement structures as homogeneous spaces.

Main Methods:

• Utilizing order topology on the base set to define a topology on the automorphism group.
• Analyzing the topological action of the automorphism group on the base set and its Cartesian products.
• Deriving local compactness for connected order topologies.

Main Results:

• The automorphism group is proven to be a topological group.
• The automorphism group acts topologically on the base set and its rank-ordered Cartesian products.
• Local compactness of the automorphism group is established for connected order topologies.
• Generalization of the Alper-Narens theorem for the case N=M.

Conclusions:

• The findings provide a deeper understanding of the structure and properties of measurement scales.
• The results establish a connection between algebraic properties (automorphism group) and topological properties (order topology).
• This work extends existing theorems in measurement theory and offers new interpretations of homogeneous spaces.