Internal structure and analytical representation of preference relations defined on infinite-dimensional real vector spaces

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.