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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

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.

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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

1