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Partial Orderings

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 Orderings

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  1. Partial Orderings

  2. 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.

  3. 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

  4. 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)

  5. 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

  6. 1 2 3 1 2 3 1 2 3 3 2 1 3 2 1 Constructing Hasse Diagrams • Example: Construct the Hasse diagram for ({1,2,3},)

  7. 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

  8. 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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