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# Partial Orderings - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Partial Orderings' - xiang

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Presentation Transcript

### 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.

• 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

• 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

• Graphical representation of a poset

• It eliminates all implied edges (reflexive, transitive)

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

• Algorithm:

• 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

2 3

1

2 3

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2 3

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Constructing Hasse Diagrams

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

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

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