same set in the partition, and otherwise inherit the order of the sets that contain them. Representation by functions For sets of sufficiently small cardinality, a third axiomatization is possible, based on real-valued functions. If is any set then a real-valued function on induces a strict weak order on by setting The associated total preorder is given by setting and the associated equivalence by setting The relations do not change when is replaced by (composite function), where is a strictly increasing real-valued function defined on at least the range of Thus for example, a utility function defines a preference relation. In this context, weak orderings are also known as preferential arrangements. If is finite or countable, every weak order on can be represented by a function in this way. However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the lexicographic order on Thus, while in most preference relation models the relation defines a utility function up to order-preserving transformations, there is no such function for lexicographic preferences. More generally, if is a set, is a set with a strict weak ordering and is a function, then induces a strict weak ordering on by setting As before, the associated total preorder is given by setting and the associated equivalence by setting It is not assumed here that is an injective function, so a class of two equivalent elements on may induce a larger class of equivalent elements on Also, is not assumed to be a surjective function, so a class of equivalent elements on may induce a smaller or empty class on However, the function induces an injective function that maps the partition on to that on Thus, in the case of finite partitions, the number of classes in is less than or equal to the number of classes on Related types of ordering Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability. A strict weak order that is trichotomous is called a strict total order. The total preorder which is the inverse of its complement is in this case a total order. For a strict weak order another associated reflexive relation is its reflexive closure, a (non-strict) partial order The two associated reflexive relations differ with regard to different and for which neither nor : in the total preorder corresponding to a strict weak order we