Partial Orderings in Mathematics
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
Constructing Hasse Diagrams
Example
: Construct the Hasse diagram for ({1,2,3},)
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
b
S
a
is the
least
element 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
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 bound
of
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
The concepts of partial orderings, comparable/incomparable elements, partial/total orders, Hasse diagrams, constructing Hasse diagrams, and maximal/minimal elements in posets. Dive into the foundations of posets and their graphical representations.
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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.
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
Constructing Hasse Diagrams Example: Construct the Hassediagram for ({1,2,3}, ) 1 1 1 2 3 2 3 2 3 3 3 2 2 1 1
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 h j 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 h j 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 g f d e b c a
