Abstract
The scale type of a measurement structure, i.e., an ordered set with a series of relations defined on that set, is described by the degree of homogeneity, M, and uniqueness, N, of its automorphism group. In the present paper the case 1⩽M=N<∞ is considered. The automorphism group is shown to be a topological group. The topology derives from the order topology on the base set. Furthermore, the automorphism group acts topologically on the base set A of the structure and on its rank-ordered Cartesian products. This result leads to an interpretation of these sets as homogeneous spaces. Subsequently, for connected order topologies the property of local compactness of the automorphism group is derived and related results proved. The famous theorem of Alper and Narens on the characterization of possible finite scale types on the reals (postulating 1⩽M⩽N⩽2) is generalized in the case N=M.
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