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

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. We suppose that the internal structure of a compatible partial preference is elementary if its positive cone is relatively algebraic open, and we refer to such compatible partial preferences as relatively open ones. It is proved that an arbitrary compatible partial preference is the disjoint union of the partially ordered family of relatively open compatible partial preferences and, moreover, as an ordered set this family is an upper semilattice. We identify the structure of this upper semilattice with the internal structure of a compatible partial preference. Further we prove that each compatible weak preference is analytically represented by a step-linear function while for the analytical representation of a compatible partial preference one needs the family of step-linear functions.