Upload
vishal-gaur
View
235
Download
0
Embed Size (px)
Citation preview
8/6/2019 7 Lattices
1/25
a b
a b
a b
8/6/2019 7 Lattices
2/25
Introduction
A relation Ron a set Sis called a partialordering orpartial orderif it is:
reflexive
antisymmetric transitive
A set Stogether with a partial ordering Riscalled apartially ordered set, orposet, and is
denoted by (S,R).
8/6/2019 7 Lattices
3/25
8/6/2019 7 Lattices
4/25
Example
Is the u relation is a partial ordering on the set
of integers?
Since a u a for every integera, u is reflexive
Ifa u b and b u a, then a = b. Hence u is anti-
symmetric.
Since a u b and b u c implies a u c, u is
transitive.
Therefore u is a partial ordering on the integers
and (Z, u) is a poset.
8/6/2019 7 Lattices
5/25
Comparable/Incomparable
The elements a and b of a poset (S, R) are called
comparable if eitheraRb orbRa.
The elements a and b of a poset (S, R) are calledincomparable if neitheraRb norbRa.
In the poset (Z+, /):
Are 3 and 9 comparable?
Are 5 and 7 comparable?
8/6/2019 7 Lattices
6/25
Totally Ordered
If every two elements of a poset (S, R) are
comparable, then Sis called a totally orderedorlinearly orderedset and R is called a total
orderorlinear order.
The poset (Z+, e) is totally ordered. Why?
The poset (Z+, /) is not totally ordered. Why?
8/6/2019 7 Lattices
7/25
Hasse Diagram
Graphical representation of a poset
Since a poset is by definition reflexive and
transitive and antisymmetric, the graphical
representation for a poset can be compacted.
Why do we need to include loops at every
vertex? Since its a poset, it must have loops
there.
8/6/2019 7 Lattices
8/25
Constructing a Hasse Diagram
Start with the digraph of the partial order.
Remove the loops at each vertex.
Remove all edges that must be presentbecause of the transitivity.
Arrange each edge so that all arrows point
up.
Remove all arrowheads.
8/6/2019 7 Lattices
9/25
Example
Construct the Hasse diagram for ({1,2,3},e)
1
2 3
1
2 3
1
2 3
3
2
1
3
2
1
Hasse DiagramThe Hasse Diagram of a finite poset S is the directed
graph whose vertices are the elements of S and there isa directed edge from a to b whenever a
8/6/2019 7 Lattices
10/25
8/6/2019 7 Lattices
11/25
Hasse Diagram LetA be a subset of (S, R).
IfuSsuch that aRu for all aA, then u is called an upperboundofA.
IflSsuch that lRa for all aA, then lis called an lowerboundofA.
Ifxis an upper bound ofA andxRzwheneverzis an upperbound ofA, thenxis called the leastupper bound(supremum) ofA.
Ifyis a lower bound ofA and zRywheneverzis a lowerbound ofA, then yis called the greatest lower bound
(infimum) ofA.
8/6/2019 7 Lattices
12/25
Exampleh j
g f
d e
b c
a
Maximal elements: h, jMinimal elements: aGreatest element: noneLeast element: aUpper bound of {a,b,c}: {e, f, j, h}Least upper bound of {a,b,c}: eLower bound of {a,b,c}: {a}Greatest lower bound of {a,b,c}: a
8/6/2019 7 Lattices
13/25
Lattices
A partially ordered set in which every pairof elements has both a least upper boundand greatest lower bound is called alattice.
f
e
c d
b
a
h
e f g
b c d
a
8/6/2019 7 Lattices
14/25
8/6/2019 7 Lattices
15/25
Let P(S) be a collection
of sets closed under
intersection and union.
Then (P(S), , ) is a
Lattice. In this Latticepartial ordering relation
is the set inclusion.
8/6/2019 7 Lattices
16/25
EXAMPLE
For any positive integer m, we will let
denote the set of divisors of m ordered bydivisibility. Then find the Hasse Diagram of
mD
}36,18,12,9,6,4,3,2,1{36 !D
1
2
6
3
12 18
36
49 Inf(a,b) = gcd(a,b) and
sup(a,b) = lcm(a,b) exist forany pair a,b in
mD
8/6/2019 7 Lattices
17/25
Another definitionofa Lattice
Let L be a non-empty set closed under two binaryoperations called meet ( ) and join ( ), where
ab=Inf.(a, b) and ab=Sup.(a, b). Then L is
called a lattice if the following axioms hold where
a, b, c are elements in L:
Idempotent: a a = a, a a = a
Commutative: a b = b a, a b = b a
A
ssociative: (a
b)
c = a
(b
c),(a b) c = a (b c)
Absorption: (a b) a = a, (a b) a = a
8/6/2019 7 Lattices
18/25
Sublattice
Suppose M is a nonempty subset of a lattice L. We
say M is sublattice of L if M itself is a lattice (with
respect to operations ofL). Thus M is a sublattice of
L iff M is closed under the operations of and
ofL. For example, the set Dm of divisors of m is a
sublattice of the positive integers N under divisibility.
8/6/2019 7 Lattices
19/25
IsomorphicLattices
Two lattices L and L are said to be isomorphic if
there is a 1-1 correspondence
f: Lp L such that
f(ab) = f(a) f(b)
& f(ab) = f(a) f(b)
for all elements a, b in L.
8/6/2019 7 Lattices
20/25
Bounded lattices
A lattice L is said to have a lower bound 0 if for anyelement x in L we have . Similarly, L is said tohave an upper bound1 if for any x in L we have .We say L is bounded ifL has both a lower bound 0 andan upper bound 1. In such a lattice we have theidentities
for any element a in L.
Ex.The lattice P(U) of all subsets
of any universal set U is a bounded
lattice with U as an upper bound
and the empty set as a lower bound.
xe0
1ex
00,0,1,11 !!!! aaaaaa1
0
x z
8/6/2019 7 Lattices
21/25
Distributive lattices
A lattice L is said to be distributive if for any
elements a, b, c in L we have the following:
a (b c) = (a b) (a c) &a (b c) = (a b) (a c)
Otherwise, L is said to be nondistributive.
Theorem:A lattice L is nondistributive
iff it contains a sublatticeisomorphic to figures on the
R.H.S.
0
yx
z
1
a c
0
b
1
8/6/2019 7 Lattices
22/25
Join irreducible elements, atoms
Let L be a lattice with a lower bound 0.
An element a in L is said to join irreducible
ifa = x y implies a = x or a = y.
Clearly 0 is join irreducible. If a has at leasttwo immediate predecessors, say b1 and b2,then a = b1 b2 , and so a is not joinirreducible. On the other hand, if a has aunique immediate predecessor c, then a isjoin irreducible.
Those elements which immediately succeed
0, are called atoms.
8/6/2019 7 Lattices
23/25
Complements
Let L be a bounded lattice with lower bound 0 and
upper bound 1. Let a be an element of L. An
element x in L is called a complement of a if
a x = 1 and a x = 0
Note: Complements need not exist and need not
be unique. Clearly y is a complement of x and z
and x and z are complements of y.
0
x
zy
1
Theorem : Let Lbe a bounded
distributive lattice. Then
complements are unique if they exist.
8/6/2019 7 Lattices
24/25
ComplementedLatticeA lattice is saidto be complemented lattice ifL
is bounded and every elementinLhas a
complement
x
z
y
1
0
This is a complemented lattice where
complements are notunique.
The lattice (P(S), )is complementedand each subsetAofShas unique
ComplementAc= S-A.
8/6/2019 7 Lattices
25/25
D70={1,2,5,7,10,14,35,70} 70
14 1035
572
1