Partial Orders

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Partial Orders
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Definition
A relation R on a set S is a partial ordering if
it is reflexive, antisymmetric, and transitive
A set S with a partial ordering R is called a
partially ordered set or a poset and is denoted
(S,R)
It is a partial ordering because pairs of
elements may be incomparable!
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A poset has no cycles
Proof
Assume a poset (S,R) does have a cycle
(a,b),(b,c),(c,a)

• A poset is reflexive, antisymmetric, and
  transitive
• (a,b) is in R and (b,c) is in R
• consequently (a,c) is in R, due to transitivity
• (c,a) is in R, by our assumption above
• (c,a) is in R and (a,c) is in R
• this is symmetric, and contradicts our
  assumption
• consequently the poset (S,R) cannot have a cycle

What kind of proof was this?
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Example

• show it is
• reflexive
• antisymmetric
• transitive

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(No Transcript)
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The equations editor has let me down
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Definition
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Example

• 3 and 9 are comparable
• 3 divides 9
• 5 and 7 are incomparable
• 5 does not divide 7
• 7 does not divide 5

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Example
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Definition
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Example

• reflexive
• antisymmetric
• transitive
• totally ordered
• all pairs are comparable
• every subset has a least element
• note Z rather than Z

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Read

• about lexicographic ordering
• pages 417 and 418

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Hasse Diagrams

• A poset can be drawn as a digraph
• it has loops at nodes (reflexive)
• it has directed asymmetric edges
• it has transitive edges
• Draw this removing all redundant information
• a Hasse diagram

• remove all loops
• (x,x)
• remove all transitive edges
• if (x,y) and (y,z) remove (x,z)
• remove all direction
• draw pointing upwards

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Example of a Hasse Diagram
The digraph of the above poset (divides)
has loops and an edge (x,y) if x divides y
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Example of a Hasse Diagram
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Exercise of a Hasse Diagram

• Draw the Hasse diagram for the above poset
• consider its digraph
• remove loops
• remove transitive edges
• remove direction
• point upwards

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Exercise of a Hasse Diagram
(5,5),(5,4),(5,3),(5,2),(5,1),(5,1),
(4,4),(4,3),(4,2),(4,1),(4,0),
(3,3),(3,2),(3,1),(3,0), (2,2),(2,1),(2,0),
(1,1),(1,0), (0,0)
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Maximal and Minimal Elements

• Maximal elements are at the top of the Hasse
  diagram
• Minimal elements are at the bottom of the Hasse
  diagram

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Example of Maximal and Minimal Elements

• Maximal set is 8,12,9,10,7,11
• Minimal set is 1

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Greatest and Least Elements

• There is no greatest Element
• The least element is 1

Note difference between maximal/minimal and
greatest/least
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Lattices
Read pages 423-425
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Topological Sorting
Read pages 425-427
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fin