  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.
