Symmetry,scale types,and generalizations of classical physical measurement

A review is presented of a number of recent results concerning fundamental measurement structures with a particular emphasis on generalizations of physical measurement. Relational structures are classified in terms of richness and redundancy of their automorphism groups (i.e., in terms of their symmetries). By means of this classification, the possible types of measurement scales are described, and the possibilities for structures with concatenation operations (either associative or nonassociative and either positive or intensive) and for conjoint structures whose automorphisms factor into component automorphisms are explicitly spelled out. Certain conjoint structures with concatenation operations on components are viewed as generalizations of classical physical measurement and some of their most important algebraic and representational properties are explored.