get and while in the partial order given by the reflexive closure we get neither nor For strict total orders these two associated reflexive relations are the same: the corresponding (non-strict) total order. The reflexive closure of a strict weak ordering is a type of series-parallel partial order. All weak orders on a finite set Combinatorial enumeration The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an -element set is given by the following sequence : These numbers are also called the Fubini numbers or ordered Bell numbers. For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are three ways of partitioning the items into one singleton set and one group of two tied items, and each of these partitions gives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering is reversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons, which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items. Adjacency structure Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves that add or remove a single order relation to or from a given ordering. For instance, for three elements, the ordering in which all three elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strict weak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weak orderings on a set are more highly connected. Define a to be a weak ordering with two equivalence classes, and define a dichotomy to be with a given weak ordering if every two elements that are related in the ordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as a Dedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies. For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of moves that add or remove one