PartialOrder

A partial order <= on a SeT X is a MathematicalRelation satisfying the following conditions: for every a,b,c in X,

    a <= a                              (reflexive property)
    If a <= b, b <= a, then a = b       (antisymmetric property)
    If a <= b, b <= c, then a <= c      (transitive property)

A set with a partial order on it is called a partial ordered set, or poset. A poset where any two elements have both a greatest lower bound and a least upper bound forms an algebraic structure called a LatticE. Every poset (X,<=) has a unique dual poset (X,>=).

Given any a,b in X, we define the OpenInterval? (a,b) = {x in X : a < x < b}. Arbitrary unions of OpenInterval?s define a TopOlogy on the poset, called the OrderTopology?. Most familiar topologies are built up from objects of this sort, e.g. the RealNumbers?.

Examples of partial orders include implications and inclusions ("is a subset of" and the more general "is a subobject of").