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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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Constructing Hasse Diagrams
  • Example: Construct the Hasse diagram for ({1,2,3},)
maximal and minimal elements

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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

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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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