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