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## PowerPoint Slideshow about ' Partial Orderings' - xiang

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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

- 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

- 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:
- 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

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},)

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 bS such that a≼b
- a is minimal in (S, ≼) if there is no bS such that b≼a
- a is the greatest elementof (S, ≼) if b≼a for all bS
- a is the leastelement of (S, ≼) if a≼b for all bS
- 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 uS such that a≼u for all aA, 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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