A Theorem on Order Extensions: Embeddability of a System of Weak Orders to Meet Solvability Constraints

Abstract

In this note an embeddability theorem is proved. It is shown that a system (A, ≤, L, M), in which A is a set and ≤, L, M are weak orders on A fulfilling a few compatibility constraints, can be extended to a larger system satisfying certain solvability conditions. The original structure can be shown to be closely related to conjoint measurement of independent components. Since in the extended structure the solvability of equations which are needed to apply representation theorems of measurement theory is guaranteed, the main result of this paper adds to the applicability of conjoint structures. The proof uses a well known extension theorem of order theory by Szpilrajn.