PartialOrderedSet

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)

PartialOrderedSets ("posets") where every two elements have both a GreatestLowerBound and a LeastUpperBound are called LatticEs, and include things like sets, groups, or the like under the partial ordering of inclusion. Posets which satisfy trichotomy (for all a,b, a<=b or b<=a) are called TotalOrderedSets, and include sets of OrdinalNumbers.

We write a < b if a <= b and the two are not equal. Given any two a,b in X, the OpenInterval (a,b) is defined to be the set of all x in X such that a < x < b. Arbitrary unions of OpenIntervals define the order-TopOlogy of the poset. Most familiar topologies are built up from objects of this sort, e.g. the RealNumbers.

A map f:X->Y from one poset to another is called an order-HomoMorphism if f(x)<=f(y) when x<=y; if it is bijective, it is called an order-IsoMorphism. Any order-based property a poset has will also be true in any poset isomorphic to it. Every poset (X,<=) has a unique dual poset (X,>=).