³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980276961³2ip³2129.97.182.48³2data³2newauthor³31³3minor³30³3text³3A partial order <= on a set X is a RelatioN 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)

PartialOrderedSet''''s ("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 ordinals.

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,>=).
³3summary³3³2tscreate³2980276961³2keepts³2980277007³2id³21076³2host³2cnts2p48.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980277007³2ip³2129.97.182.48³2data³2text³3A partial order <= on a set X is a RelatioN 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)

PartialOrderedSet''''s ("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 TotalOrderedSet''''s, and include sets of ordinals.

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,>=).
³3minor³31³3newauthor³30³3summary³3³2tscreate³2980276961³2keepts³2980277034³2id³21076³2host³2cnts2p48.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980277034³2ip³2129.97.182.48³2data³2newauthor³30³3minor³31³3text³3A partial order <= on a set X is a RelatioN 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)

PartialOrderedSet''''''s ("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 TotalOrderedSet''''''s, and include sets of ordinals.

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,>=).
³3summary³3³2tscreate³2980276961³2keepts³2980277075³2id³21076³2host³2cnts2p48.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980277075³2ip³2129.97.182.48³2data³2text³3A partial order <= on a set X is a RelatioN 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)

PartialOrderedSet''''s ("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 TotalOrderedSet''''s, and include sets of ordinals.

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,>=).
³3minor³31³3newauthor³30³3summary³3³2tscreate³2980276961³2keepts³2980277089³2id³21076³2host³2cnts2p48.uwaterloo.ca³2revision³24³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980277089³2ip³2129.97.182.48³2data³2newauthor³30³3minor³31³3text³3A partial order <= on a set X is a RelatioN 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)

PartialOrderedSet'''''s ("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 TotalOrderedSet'''''s, and include sets of ordinals.

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,>=).
³3summary³3³2tscreate³2980276961³2keepts³2980277097³2id³21076³2host³2cnts2p48.uwaterloo.ca³2revision³25³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980277097³2ip³2129.97.182.48³2data³2text³3A partial order <= on a set X is a RelatioN 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)

PartialOrderedSet''''''s ("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 TotalOrderedSet''''''s, and include sets of ordinals.

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,>=).
³3minor³31³3newauthor³30³3summary³3³2tscreate³2980276961³2keepts³2980277122³2id³21076³2host³2cnts2p48.uwaterloo.ca³2revision³26³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980277122³2ip³2129.97.182.48³2data³2newauthor³30³3minor³31³3text³3A partial order <= on a set X is a RelatioN 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)

PartialOrderedSet''s'' ("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 TotalOrderedSet''s'', and include sets of ordinals.

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,>=).
³3summary³3³2tscreate³2980276961³2keepts³2980277131³2id³21076³2host³2cnts2p48.uwaterloo.ca³2revision³27³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980277131³2ip³2129.97.182.48³2data³2text³3A partial order <= on a set X is a RelatioN 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)

PartialOrderedSet''s'' ("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 TotalOrderedSet''s'', 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,>=).
³3minor³31³3newauthor³30³3summary³3³2tscreate³2980276961³2keepts³2980402112³2id³21076³2host³2cnts2p48.uwaterloo.ca³2revision³28³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980402112³2ip³2129.97.182.64³2data³2newauthor³30³3minor³31³3text³3A 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)

PartialOrderedSet''s'' ("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 TotalOrderedSet''s'', 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,>=).
³3summary³3*³2tscreate³2980276961³2keepts³2980496412³2id³21076³2host³2cnts2p64.uwaterloo.ca³2revision³29