either () or () or ( is incomparable with and is incomparable with ). Total preorders Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the same mathematical concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total preorder or weak order is a preorder in which any two elements are comparable. A total preorder satisfies the following properties: : For all if then : For all Which implies : for all A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preorders are sometimes also called preference relations. The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we take the converse of the complement: for a strict weak ordering define a total preorder by setting whenever it is not the case that In the other direction, to define a strict weak ordering < from a total preorder set whenever it is not the case that In any preorder there is a corresponding equivalence relation where two elements and are defined as equivalent if In the case of a preorder the corresponding partial order on the set of equivalence classes is a total order. Two elements are equivalent in a total preorder if and only if they are incomparable in the corresponding strict weak ordering. Ordered partitions A partition of a set is a family of non-empty disjoint subsets of that have as their union. A partition, together with a total order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition and by Theodore Motzkin a list of sets. An ordered partition of a finite set may be written as a finite sequence of the sets in the partition: for instance, the three ordered partitions of the set are In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partition gives rise to a strict weak ordering in which two elements are incomparable when they belong to the