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

    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