³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980496212³2ip³2129.97.130.42³2data³2newauthor³31³3minor³31³3text³3A 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.

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.
³3summary³3*³2tscreate³2980496212³2keepts³2980496309³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980496309³2ip³2129.97.130.42³2data³2text³3A 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").
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980496212³2keepts³2980496347³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980496347³2ip³2129.97.130.42³2data³2newauthor³30³3minor³31³3text³3A 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").
³3summary³3*³2tscreate³2980496212³2keepts³2980730035³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³23