Preorder

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or ) preorder is a partial order, and a symmetric preorder is an equivalence relation. The name comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol can be used as the notational device for the relation. However, because they are not necessarily antisymmetric, some of the ordinary intuition associated to the symbol may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied. In words, when one may say that b a or that a b, or that b to a. Occasionally, the notation ← or → or is used instead of To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. In general, the corresponding graphs may contain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components. Consider a homogeneous relation on some given set so that by definition, is some subset of and the notation is used in place of Then is called a or if it is reflexive and transitive; that is, if it satisfies: Reflexivity: for all and Transitivity: if for all A set that is equipped with a preorder is called a preordered set (or proset).