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 ID: 315169
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Slide1
Partial OrderingsSlide2
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
).
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
.Slide3
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
⫮
5Slide4
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) Slide5
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 arrowheadsSlide6
Constructing
Hasse
Diagrams
Example
: Construct the
Hasse
diagram for ({1,2,3},
)
1
2 3
1
2 3
1
2 3
3
2
1
3
2
1Slide7
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 element of (S,
≼) if b≼a for all bS
a is the least element of (S, ≼) if
a≼
b for all b
Sg
reatest 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:
aSlide8
Upper and
L
ower Bounds
Let
A
be a
subset
of (
S
,
≼
)
If
u
S 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 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