Nondecomposable conjoint measurement for bisymmetric structures

This paper discusses two “nondecomposable” conjoint measurement representations for an asymmetric binary relation ? on a product set A × X, namely (a, x) ? (b, y) iff f₁(a) + g₁(a)g₂(x) > f₁(b) + g₁(b)g₂(y), and (a, x) ? (b, y) iff f₁(a) + f₂(x) + g₁(a)g₂(x) > f₁(b) + f₂(y) + g₁(b)g₂(y). Difficulties in developing axioms for ? on A × X which imply these representations in a general formulation have led to their examination from the standpoint of bisymmetric structures based on applications of a binary operation to A × X. Depending on context, the binary operation may refer to concatenation, extensive or intensive averaging, gambles based on an uncertain chance event, or to some other interpretable process. Independence axioms which are necessary and sufficient for the special representations within the context of bisymmetric structures are presented.