# 7 Lattices

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a b

a b

a b

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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).

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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.

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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?

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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?

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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.

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Constructing a Hasse Diagram

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.

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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

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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.

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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

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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

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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.

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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

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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

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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.

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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.

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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

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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

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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.

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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.

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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.

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D70={1,2,5,7,10,14,35,70} 70

14 1035

572

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