Partial orderings
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Partial Orderings. Partial Orderings. A relation R on a set S is called a partial ordering if it is: r eflexive antisymmetric transitive A set S together with a partial ordering R is called a partially ordered set , or poset , and is denoted by ( S , R ).

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Partial orderings1
Partial Orderings

  • A relation R on a set S is called a partial ordering if it is:

    • reflexive

    • antisymmetric

    • transitive

  • A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S,R).

  • Example: “” is a partial ordering on the set of integers

    • reflexive: a  a for every integer a

    • anti-symmetric: If a  b and b  a then a = b

    • transitive: a  b and b  c implies a  c

    • Therefore “” is a partial ordering on the set of integers and (Z, ) is a poset.


Comparable incomparable elements
Comparable/Incomparable Elements

  • Let “≼” denote any relation in a poset (e.g. )

  • The elements a and b of a poset (S, ≼) are:

    • comparable if either a≼b or b≼a

    • incomparable if neither a≼b nor b≼a

  • Example: Consider the poset (Z+,│), where “a│b” denotes “a divides b”

    • 3 and 9 are comparable because 3│9

    • 5 and 7 are not comparable because nether 5⫮7 nor 7⫮5


Partial and total orders
Partial and Total Orders

  • If some elements in a poset(S, ≼) are incomparable, then it is partially ordered

    • ≼ is a partial order

  • If every two elements of a poset (S, ≼) are comparable, then it is totally ordered or linearly ordered

    • ≼ is a total (or linear) order

  • Examples:

    • (Z+,│) is not totally ordered because some integers are incomparable

    • (Z, ≤) is totally ordered because any two integers are comparable (a ≤ b or b ≤ a)


Hasse diagrams
Hasse Diagrams

  • Graphical representation of a poset

    • It eliminates all implied edges (reflexive, transitive)

    • Arranges all edges to point up (implied arrow heads)

  • Algorithm:

    • Start with the digraph of the partial order

    • Remove the loops at each vertex (reflexive)

    • Remove all edges that must be present because of the transitivity

    • Arrange each edge so that all arrows point up

    • Remove all arrowheads


Constructing hasse diagrams

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2 3

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2 3

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2 3

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1

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Constructing Hasse Diagrams

  • Example: Construct the Hasse diagram for ({1,2,3},)


Maximal and minimal elements

h j

g f

d e

b c

a

Maximal and minimal Elements

  • Let (S, ≼) be a poset

  • a is maximal in (S, ≼) if there is no bS such that a≼b

  • a is minimal in (S, ≼) if there is no bS such that b≼a

  • a is the greatest elementof (S, ≼) if b≼a for all bS

  • a is the leastelement of (S, ≼) if a≼b for all bS

    • greatest and least must be unique

  • Example:

  • Maximal: h,j

  • Minimal: a

  • Greatest element: None

  • Least element: a


Upper and l ower bounds

h j

g f

d e

b c

a

Upper and Lower Bounds

  • Let A be a subset of (S, ≼)

  • If uS such that a≼u for all aA, then u is an upper bound of A

  • If x is an upper bound of A and x≼z whenever z is an upper bound of A, then x is the least upper boundof A (must be unique)

  • Analogous for lower bound and greatest upper bound

  • Example: let A be {a,b,c}

  • Upper bounds of A: e,f,j,h

  • Least upper bound of A: e

  • Lower bound of A: a

  • Greatest lower bound of A: a


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