partial orderings
DESCRIPTION
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 ). - PowerPoint PPT PresentationTRANSCRIPT
Partial Orderings
Partial OrderingsA relation R on a set S is called a partial
ordering if it is:reflexiveantisymmetrictransitive
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 integersreflexive: a a for every integer aanti-symmetric: If a b and b a then a = btransitive: a b and b c implies a cTherefore “” is a partial ordering on the set of
integers and (Z, ) is a poset.
Comparable/Incomparable ElementsLet “≼” 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≼aincomparable 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│95 and 7 are not comparable because nether 5 7⫮
nor 7 5⫮
Partial and Total OrdersIf 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 DiagramsGraphical 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 orderRemove the loops at each vertex (reflexive)Remove all edges that must be present
because of the transitivityArrange each edge so that all arrows point upRemove all arrowheads
Constructing Hasse DiagramsExample: Construct the Hasse diagram for
({1,2,3},) 1
2 3
1
2 3
1
2 3
3
2
1
3
2
1
Maximal and minimal ElementsLet (S, ≼) be a poseta 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 element of (S, ≼) if b≼a for all
bSa is the least element of (S, ≼) if a≼b for all bS
greatest and least must be unique h j
g f
d e
b c
a
Example:• Maximal: h,j• Minimal: a• Greatest element:
None• Least element: a
Upper and Lower BoundsLet A be a subset of (S, ≼)If uS such that a≼u for all aA, then u is an
upper bound of AIf x is an upper bound of A and x≼z whenever z is
an upper bound of A, then x is the least upper bound of A (must be unique)
Analogous for lower bound and greatest upper bound h j
g f
d e
b c
a
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