Internal structure and analytical representation of preference relations defined on infinite-dimensional real vector spaces
V. V. Gorokhovik
TL;DR
The paper extends the theory of compatible preference relations to infinite-dimensional real vector spaces by linking orders to positive cones and intrinsic cores. It develops a detailed internal structure, showing relatively open components form an upper lattice for partial preferences and a linear-chain refinement for weak preferences, with monotone linear functionals attached to components. It then establishes an analytical representation of compatible weak preferences via step-linear functions and extends partial preferences to weak ones (Szpilrajn-type) while preserving analyticity through separation results. Collectively, these results provide a rigorous framework for representing and extending preferences in infinite-dimensional vector spaces, with implications for vector optimization and decision analysis.
Abstract
The paper deals with partial and weak preference relations defined on infinite-dimensional vector spaces and compatible with algebraic operations. By a partial preference we mean an asymmetric and transitive binary relation, while a weak preference is such a partial preference for which the indifference relation corresponding it is transitive (an indifference relation is the complement to the union of a partial preference and the reverse to it). Our first aim is to study the internal structure of compatible partial and weak preferences. Using these results we then prove that a compatible weak preference admits an analytical representation by means of a step-linear function, while a compatible partial preference can be analytically represent by the family of step-linear functions.