Generalized Degrees and Densities for Families of Sets

Home , Family of sets

GENERALIZED DEGREES AND

DENSITIES FOR FAMILIES OF SETS

EMANUEL KNILL

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulllment

of the requirements for the degree of

Do ctor of Philosophy

Department of Mathematics

This thesis for the Do ctor of Philosophy degree by

Emanuel Knill

has b een approved for the

Department of

Mathematics

Richard Laver

Andrzej Ehrenfeucht Date

Knill Emanuel Ph D Mathematics

Generalized Degrees and Densities for Families of Sets

Thesis directed by Richard Laver

Let F b e a family of subsets of f ng The widthdegree of an

element x F is the width of the family fU F jx U g If F has maximum

widthdegree at most k then F is locally k wide

Bounds on the size of lo cally k wide families of sets are established If

F is lo cally k wide and centered every U F has an element which do es not

b elong to any member of F incomparable to U then F k n this

b ound is b est p ossible Nearly exact b ounds linear in n and k on the size of

lo cally k wide families of arcs or segments are determined If F is any lo cally

F is linearly b ounded in n The pro of of this result k wide family of sets then

involves an analysis of the combinatorics of antichains

Let P b e a p oset and L a semilattice or an intersectionclosed family

of sets The P size of L is L For u L the P density of u is the ratio

The density of u is given by the density of u Let p b e the number of

lters of P L has the P density property i there is a joinirreducible a L

such that the P density of a is at most

Which nontrivial semilattices have the P density prop erty For P

it has b een conjectured that the answer is all the unionclosed sets

conjecture Certain sub direct pro ducts of lowersemimodular lattices and for

P n of geometric lattices have the P density prop erty in a strong sense This

generalizes some previously known results A xed lattice has the ndensity

prop erty if n is large enough The density of a generator U of a unionclosed

family of sets L with L is estimated The estimate dep ends only on the lo cal

prop erties of L at U If L is generated by sets of size at most two then there is

a generator U of L with estimated density at most

CONTENTS

CHAPTER

INTRODUCTION

To the Reader

Overview

PRELIMINARIES

Conventions and Notation

Sets Arcs and Segments

Families of Sets

Partially Ordered Sets

Lattices and Semilattices

Two Results from the Combinatorics of Sets

Notes

FAMILIES OF SETS WITH WIDTH RESTRICTIONS

Denitions

The Structure of Centered Families of Sets

Basic Extremal Results

The Size of k pseudotrees

Intersecting Antichains of Arcs

The Size of Lo cally k wide Families of Arcs and Segments

Maximal r antichains and Sp erner Closure

The Size of Lo cally k wide Families of Sets

Notes

POSET DENSITIES IN SEMILATTICES

Introduction

Examples and Counterexamples

Semilattices Intersection and Unionclosed Families of Sets

Sub direct Pro ducts of Semilattices

Constructions which Preserve Matching Prop erties

Neighborho o ds in Lattices

Lattices with Sp ecial Lattice Neighborho o ds

The Density Perspective

The ndensity Prop erty for Large n

A Bound on the Minimum P density of Joinirreducibl es

Unionclosed Families of Sets Generated by Graphs

Notes

BIBLIOGRAPHY

CHAPTER

INTRODUCTION

To the Reader

Section is an overview of the contents of this work

Chapter is a summary of the background material and contains def

initions and some fundamental results from the combinatorics of nite sets and

the theory of p osets Most of the denitions and terminology are standard

Table in Section contains a list of notation and terminology

Except in Chapters and most denitions app ear after the paragraph

heading Denition Terms are italicized when rst dened Observations are

statements which are true by denition or follow from the discussion and do not

require further pro of

Chapters and are indep endent and can b e read in either order

The Notes at the end of each chapter contain a discussion of the liter

ature relevant to each section attributions of results and references to related

work

Overview

The study of extremal prop erties of families of sets involves determining

the relationships b etween their various size measures Two fundamental size

measures of a family of sets F are F the cardinality of the family and F

the number of elements in the union of its members Many size measures are

obtained by considering F as a p oset ordered by inclusion One such measure

is the width of the family F Another measure is the P size of F given by the

number of orderpreserving maps from the p oset P into F For example if P is

the nelement chain then the P size of F is the number of multichains of length

n of F

Let w b e a size measure for families of sets Let F fU F j x U g

The domain of F is the set F The measure w induces a generalized degree

measure as follows If x is an element of the domain of F then the w degree

of x is w F If F is a graph ie every member of F is a pair and w is the

cardinality function then the w degree coincides with the usual notion of degree

for graphs

The size measure w also induces a generalized notion of density If G

w G

is a subfamily of F then the w density in F of G is the ratio If x is an

w F

element of the domain of F then the w density in F of x is the w density in F

of F x

Much research has b een devoted to the extremal theory of sets How

ever many generalized degrees and densities have only recently b een introduced

There are some well known op en problems concerning these notions in particular

concerning the notion of density Perhaps the most famous is the unionclosed

sets conjecture due to P Frankl Conjecture A list of others can b e found

in Section

This work consists of two parts The rst introduces the study of the

widthdegree The second examines P densities in intersection and unionclosed

families of sets ie semilattices

The widthdegree A family of sets of maximum widthdegree at

most k is called locally k wide Except for Knill et al there has b een

no published work on lo cally k wide families of sets The concept was rst

introduced by Ehrenfeucht and Haussler as part of the denition of pseudotrees

Their aim was to nd structures more general than trees for classifying textual

information

A tree on the set X is a lo cally wide family F of subsets of X such

that F contains X and the singletons of X Since F is lo cally wide i every

pair of members of F is either comparable or disjoint a tree on X has the

familiar treelike structure where X is the ro ot and the singletons of X are the

leaves A tree F also has the prop erty that every U F has an element not

contained in any member of F incomparable to U Families of sets with this

prop erty are called centered Thus centered lo cally k wide families of sets are a

natural generalization of trees Such families are called k pseudotrees

In order for k pseudotrees to b e useful from a data structure p ersp ec

tive they should not b e to o large relative to the size of the domain Ehren

feucht and Haussler asked the following question Is the size of a k pseudotree

linearly b ounded in the size of the domain The sp ecial case for pseudotrees

of segments was solved by Knill in Knill and Ehrenfeucht subsequently

showed that if the domain of a k pseudotree F has n elements where n k then

F k n Theorem This b ound is b est p ossible Linear b ounds

have now b een obtained for lo cally k wide families of arcs Theorem lo

cally k wide families of segments Theorem and arbitrary lo cally k wide

families of sets Theorem The results are summarized in Table where

f k n is the maximum size of a lo cally k wide family of subsets of f ng

in the given class

The pro of of the linear b ound for lo cally k wide families of sets requires

an analysis of the combinatorics of certain antichains of p osets An antichain A

of the p oset P is a maximal r antichain i the width of the lter A generated

by A is r and A is the only antichain of size r in A A maximal antichain is a

maximal r antichain for some r The family of maximal antichains is a meet

subsemilattice of the family of antichains of P with the lter order dened by

A B i B A Theorem If A and B are incomparable antichains

then A B max A B Corollary If w is the width of P then the

w w

union of the antichains consists of at most elements Theorem

P densities Let P b e a p oset and let p b e the number of lters of

P An intersectionclosed family of sets L has the P density property i there is

The an element x in the domain of L such that the P density of x is at most

density property is the density prop erty note that for P p

The problem discussed in the second part of this work is the following

Problem What intersectionclosed families of sets have the P density

property

One version of the unionclosed sets conjecture asserts that for every

intersectionclosed family L with at least two members there is an element

in the domain of L contained in at most one half of the members of L see

Duus If true this implies that all nontrivial intersectionclosed families

have the density prop erty

If the intersectionclosed family L is trivial ie L then L do es

not have the P density prop erty for any p oset P No other examples without

the P density prop erty are known

Every intersectionclosed family is a meetsemilattice and conversely

every meetsemilattice can b e represented by an intersectionclosed family eg

the family of principal ideals of the semilattice Therefore one can use lattice

theoretical techniques to analyze Problem The notion of an element in

the domain is replaced by that of a prop er joinirreducibl e For the remainder

of this section all semilattices intersection and unionclosed families are

assumed to have at least two members

Known results related to the unionclosed sets conjecture are

i If the intersectionclosed family L contains L n fx y g for some x y

L then L has the density prop erty

ii If the average size of the members of the intersectionclosed family L is

at most L then L has the density prop erty

iii If the meetsemilattice L has the density prop erty then so do es L M

for any meetsemilattice M

iv Mo dular and geometric lattices have the density prop erty

v If L is a meetsemilattice and n is the maximum cardinality of L n a

L C n log n for some constant C for joinirreducible s a then

Results i and ii are obtained by double counting They can b e

generalized and used as a starting p oint to establish lower b ounds on counter

examples to the unionclosed sets conjecture see Sarvate and Renauld

Result iii follows from the observation that if a is a joinirreducible of L then

ha i is a joinirreducibl e of M Results iv and v are of unknown origins

They are generalized in Sections and resp ectively

To generalize iii and iv the idea of decreasing matchings among dif

ferent types of orderpreserving maps in L is introduced Let F b e a lter of P

P P

and a a joinirreducible of L Dene TL F a ff L j f x a i x F g

The orderpreserving map f P L is of type F a i f TL F a The

pair L a has the top matching property for P i for every lter F of P there

P P

is a onetoone map TL P a TL F a such that f f The

semilattice L has the top matching property i there is a joinirreducible a L

such that L a has the top matching prop erty The matching prop erty implies

the P density prop erty There are nontrivial semilattices which do not have the

matching prop erty for any p oset Example

In Section it is shown that lattices with a lowersemimodular coatom

have the matching prop erty for any p oset Theorem This includes all

lowersemimodular and in particular all mo dular lattices It is also shown that

nontrivial geometric lattices have the matching prop erty for linearly ordered

sets Theorem

The matching prop erty is preserved not only by direct pro ducts but

also by certain sub direct pro ducts which preserve lo cal prop erties of lattices

Section To make the notion of lo cality precise lattices are considered as

unionclosed families of sets Section The joinirreducibles of the lattice L

form a family of sets J which generates L by union The lattice neighborhood

N U of U J is the unionclosed family generated by fg fA J j A U

g If U J and N U is a geometric lattice then L U has the matching

prop erty for the nelement chain This and similar results app ear in Section

In Section a technique is developed for estimating the density of

a joinirreducible a of L which dep ends only on N N a This estimate is

L L

used to show that if the lattice L considered as a unionclosed family of sets is

generated by a graph then L has the density prop erty Theorem This

implies that for every graph G there is an edge contained in at most half of the

unions of edges of G

Result v listed ab ove is generalized in Section If for every join

irreducible a the number of orderpreserving maps not of type P a is at most

n then L M n where M n n log n asymptotically The b ound is

exact for a generalization of the notion of lattices to multisemilattices where

every nonmaximal member of the lattice is assigned a multiplicity

In Section an expression for the Zeta p olynomial is used to show

that if L is a xed nontrivial lattice and n is suciently large then L has the

ndensity prop erty where n is the nelement chain

Table

Class f k n Lo cation

k n

U F U r b Theorem c

n k n ln n Theorem All families

k n Theorem

k n ok n if

ok and k on Example

k n if Centered families

n k Theorem

k n k k if Families of segments

n k Theorem

k n k Theorem Families of arcs

k n k k if

n k Example

CHAPTER

PRELIMINARIES

Conventions and Notation

All sets and structures are nite by default On the few o ccasions where

an innite set is used this will b e stated explicitly

Subscripts and function arguments will b e omitted when it is p ossible

to do so without loss of clarity

The guidelines for symbol use are given in Table

Table Guidelines for symbol use

C L classes of structures

A B antichains of sets

F G H families of sets

A B C sets antichains

F G H lters

L M semilattices

P Q p osets

U V W sets members of a family of sets

X Y Z sets domains

a b c elements joinirreducible s atoms etc

i j k integers

u v w elements of a semilattice

x y z arbitrary elements real numbers

A list of notation and terminology is given in Table

Sets Arcs and Segments

An nset is a set with exactly n elements The sets A and B intersect

i A B is nonempty A and B are comparable i A B or B A A and B

are incomparable i they are not comparable A and B overlap i they intersect

and are incomparable

If i and j are integers then i j denotes the segment of integers k such

that i k j If j i then i j is the empty segment For i j the left and

right endpoints of i j are i and j resp ectively A segment of n is a segment

i j with i j n Let i j and k l b e segments Note that if k i or

Table Notation

Z the integers

N the nonnegative integers

P the strictly p ositive integers

R the real numbers

n the set f ng with the linear order

i j the set of integers k such that i k j

Z the integers mo dulo n represented by f n g

the empty set

X the number of elements cardinality size of the set X

bxc the greatest integer x

dxe the least integer x

log x the logarithm base two of x

ln x the logarithm base e of x

n the factorial of n n n

n the k th falling factorial of n

n nn n k

n n

the binomial co ecient

k k k

f g the comp osition of the maps f and g

f y A the restriction of the map f to A

f x

f og means lim

g x

f O g means that for some c f x C g x

f x

f g means lim

g x

A B the set A is a prop er subset of B

A B A B or A B

A B the intersection of the sets A and B

A B the union of the sets A and B

A n B set dierence the set of elements of A not in B

ha bi the ordered pair a and b

A B the cartesian pro duct of the sets A and B

A the cartesian pro duct of the sets A A

i n

the pro jection onto the ith comp onent of a

cartesian pro duct

d x the degree in the graph G of x G

Table continued

the family of subsets of X

F the union of the members of the family of sets F

F the intersection of the members of F

F G for families of sets F and G F G fU V j U F V G g

F the family induced by F in X F fU X j U F g

X X

F the family induced by F ab ove X

F fU X j U F g

F the restriction of F to X F fU X j U F g

X X

F the restriction of F to the complement of X

F fU n X j U F g

F the family of members of F of size or rank r

F fU F j U r g

CU the center of U in a family of sets

the least element of the p oset P if it exists

the greatest element of the p oset P if it exists

x y for x y in the p oset P the interval of elements

b etween x and y x y fz P j x z y g

A the ideal generated by A in the p oset P

A fx P j x y for some y Ag

A the lter generated by A in the p oset P

A fx P j x y for some y Ag

A the least upp er b ound of A in a p oset if it exists

A the greatest lower b ound of A in a p oset if it exists

J L the set of prop er joinirreducible s of the meetsemilattice L

M L the set of prop er meetirreducibles of the joinsemilattice L

P L the family of principal ideals of the semilattice L

Sub L the family of subsemilattices of L

P Q the disjoint union of the p osets P and Q

Q the set of orderpreserving maps from P to Q

wP the width of the p oset P

TL F a for the semilattice L p oset P lter F of P

and joinirreducible a L

P P

TL F a ff L j f x a i x F g

P Q means that P is isomorphic to Q

j l then the two segments are comparable The segment i j is to the left of

k l i i j and k l are incomparable and i k The intersection of i j and

k l is also a segment If i j is to the left of k l then their intersection is the

segment k j and is called the left overlap of i j with k l If this intersection

is nonempty then i j overlaps k l from the left The right overlap of two

segments is dened similarly

The number i precedes j in Z i j i mo dn This denes the

clockwise cyclic order on Z The number j o ccurs before k clockwise from i i

j k or j app ears b efore k in the sequence

i i mo dn i mo dn i mo dn i mo dn

In clockwise cyclic order i j k means that j o ccurs b efore k clo ckwise from

i The expression i k denotes the arc of elements j Z such that in clo ckwise

cyclic order i j k The left and right endpoints of the arc i k are i and k

resp ectively By default i k Z so that k mo dn i

The intersection of two incomparable arcs is not necessarily an arc but

consists of the union of at most two disjoint arcs The arc i j overlaps k l

from the left i k o ccurs b efore j clo ckwise from i and the arcs i j and k l

are incomparable The arc k j is called the left overlap of i j with k l

The cyclic closure of n is obtained by letting the integer n precede

in n The cyclic closure of n is identical to Z with the clo ckwise cyclic order

if k n is identied with k mo dn Z

Families of Sets

A family of sets F is a set of sets The domain of F is given by F

F is a family of arcs segments of Z n i every member of F is an arc

segment of Z n A hypergraph is a pair F X where F is a family of

sets and X includes the domain of F The members of F are the edges of the

hypergraph The elements of X not in the domain are isolated elements of the

hypergraph The hypergraph F X is r uniform i every member of F has

exactly r elements

If F and G are families of sets then F G fU V j U F V G g

and F G fU V j U F V G g The family of sets F is unionclosed i

for every U V F U V F Equivalently F is unionclosed i F F F

Similarly F is intersectionclosed i F F F

Observation Let X be a subset of the domain of the family of sets F If

F is unionclosed intersectionclosed then so are the induced families of sets

F and F as wel l as the restrictions F and F of F

X X X

Let F b e a unionclosed family of sets The family F is generated

by G i F consists of the unions of nonempty subfamilies of G that is i

F f H j H G H g The generators of F are the members U of F such

that if G F and G U then U G Note that F is generated by the family

of generators of F

A forest of sets on X is a family F of nonempty subsets of X such

that for every U V F U and V are either comparable or disjoint If in addition

F contains X and the singletons of X then F is a tree of sets

A partition of X is a family of sets F with domain X such that the

members of F are pairwise disjoint The partition G of X is coarser than F i

every member of F is included in a member of G

Partially Ordered Sets

A partially ordered set P poset for short is a nonempty set also

denoted by P together with a reexive transitive and antisymmetric binary

relation on P Thus for every x y z P x x reexivity x y and

y z imply that x z transitivity and x y and y x imply that x y

antisymmetry The relation is the partial order on P The expression x y

means x y and x y The element x is below y i x y and x is above y i

y x

Let P b e a p oset The dual partial order of P is obtained by reversing

the partial order of P The set P together with the dual partial order is denoted

by P the dual of P For x y P x y in P i y x in P Whenever

appropriate the symbols and are reversed to denote the dual partial order

If F is a family of sets the subset relation induces the inclusion order

on F dened by U V i U V When terminology for p osets is used in

the context of a family of sets the intended partial order is the inclusion order

unless otherwise sp ecied

Let P b e a p oset The elements x and y of P are comparable i x y

or y x otherwise x and y are incomparable The element y is between x and

z i x y z The element x covers y in P i y x and the only elements

b etween x and y are x and y The cover relation of P denes a directed graph

the Hasse diagram of P obtained by connecting x to y i x covers y The Hasse

diagram of P completely determines the partial order of P assuming that P is

nite Pictorial representations of the Hasse diagram where a downward edge

from x to y means that x covers y are frequently used to represent p osets

The p oset P has the discrete order i for every x y P with x y

x and y are incomparable P is a linearly ordered set or a chain i for every

x y P x y or y x Thus n is linearly ordered by the usual ordering of

the integers Figure shows the Hasse diagrams of f g with the discrete

order and

The partial order is an extension of the partial order of P i for

every x y P x y implies that x y The following theorem is one of

the fundamental results in the theory of p osets see Notes at the end of this chapter

f g

d d d

fg fg

d d d d

Figure f g with the discrete order and

Theorem For every poset P there exists a linear order which extends the

partial order of P

Let P b e a p oset If x and y are elements of P and x y then the

interval from x to y denoted by x y consists of the elements of P b etween x

and y The subset Q of P is convex i for every x and y in Q such that x y

x y Q The order induced by P on the subset Q is the restriction of the

partial order of P to Q Thus if x y Q then x y in Q i x y in P A

maximal element of Q is an element x Q such that for every y Q x y

Dually a minimal element of Q is an element x Q such that for every y Q

y x Observe that x is minimal in Q i x is maximal in Q If x is the only

maximal element of Q then x is called the greatest element of Q If x is the

only minimal element of Q then x is called the least element of Q If P has a

greatest element this element is denoted by Similarly the least element of

P if it exists is denoted by An atom is an element which covers a coatom

is an element which is covered by

The element x of P is an upper bound of Q i for every element y of Q

y x The element x is the least upper bound of Q i x is the least element of

the set of upp er b ounds of Q If it exists the least upp er b ound of Q is denoted

by Q Lower bounds and greatest lower bounds of Q are dened dually If it

exists the greatest lower b ound of Q is denoted by Q If Q fx y g then

W V

Q and Q are also denoted by x y the join of x and y and x y the meet

of x and y resp ectively To avoid ambiguity and will b e used to denote

P P

the join and the meet op erations in P when necessary

The basic prop erties of the in general partial join and meet op erations

are as follows

and are idemp otent ie x x x and x x x

and are symmetric ie x y y x if either side exists and

similarly for

and are asso ciative ie if x y y z x y z and x y z

all exist then x y z x y z and similarly for

and satisfy the absorption identities ie x y x x if x y

exists and x y x x if x y exists

If P has a greatest element then and if P has a least element

then

A proper chain of P of length n is a strictly increasing sequence

x x x of elements of P The maximum length of a chain of P

is called the height of P A multichain of length n is an increasing sequence

x x x of elements of P Note that a multichain of length n has

n elements

An antichain of P is a subset A of P such that every pair of distinct

elements of A are incomparable The maximum cardinality of an antichain of P

is called the width of P and is denoted by wP An antichain of P with wP

elements is called a Sperner antichain

If P is the disjoint union of n chains then the width of P is at most n

The converse is given by the following result due to Dilworth

Theorem If P is a poset of width w then P is the disjoint union of w

chains

An order ideal of the p oset P is a subset I of P such that if x I

and y x in P then y I An order lter of P is a subset F of P such

that if x F and y x in P then y F The family of ideals of P is b oth

intersection and unionclosed as is the family of lters of P

If A is an antichain of P and x P then x is below A i there exists

y A such that x y The set of elements of P b elow A is the ideal generated

by A and is denoted by A The element x of P is above A i there exists y A

such that x y The set of elements of P ab ove A is the lter generated by

A and is denoted by A For x P the principal ideal generated by x is

x fxg and the principal lter generated by x is x fxg

Let A b e the family of antichains of P Identifying each antichain A

with A induces a partial order on A where A B i A B This is called

the ideal order of A The lter order is the partial order of A dened by A B

i A B Note that this is the dual of the partial order induced on A by

the inclusion order of the family of lters A for A A

Let I b e an ideal of P Let A b e the set of minimal elements of I

Then A is the unique antichain which generates I Similarly if F is a lter of

P the set of maximal elements of F is the unique antichain which generates F

This establishes bijective corresp ondences b etween antichains and ideals and

b etween antichains and lters

Let F b e a family of sets partially ordered by inclusion The subfamilies

S S

of F can b e ordered by dening G H i G H for G H F This is an

extension of the inclusion order of the subfamilies of F Since for an antichain

S S

A of F A A the restriction of this order to the antichains of F extends

the ideal order Alternatively the subfamilies of F can b e ordered by dening

T T

G H i G H for G H F This extends the dual of the inclusion

T T

order of the subfamilies of F Since for an antichain A of F A A the

restriction of this order to the antichains of F extends the lter order Thus

deciding which partial order of the antichains of F to use often dep ends on

whether we are interested in intersections or in unions of antichains

Let P and Q b e p osets A map f P Q is orderpreserving i x y

in P implies that f x f y The comp osition of orderpreserving maps is

orderpreserving P and Q are isomorphic i there is an orderpreserving bijec

tion f P Q with an orderpreserving inverse The set of orderpreserving

P P

maps from P to Q is denoted by Q The set Q is partially ordered by dening

f g i for every x P f x g x in Q

Let f b e an orderpreserving map from P into the two element chain

Then the set of elements x of P such that f x is a lter of P Conversely

if F is a lter of P then the map f P dened by f x i x F is

orderpreserving This corresp ondence b etween and the set of lters of P is

order preserving

Theorem The family of lters of P is isomorphic to

Two other useful constructions on p osets are the cartesian pro duct and

the disjoint union The cartesian product P Q of P and Q has the partial order

dened by hx y i hx y i i x x and y y The disjoint union P Q of P

and Q has the partial order dened by x y i either x y P and x y in P

or x y Q and x y in Q

Lattices and Semilattices

A meetsemilattice is a p oset L such that for every u v L the greatest

lower b ound u v of u and v exists The dual of a meetsemilattice is a join

semilattice A semilattice is by default a meetsemilattice If all pairwise meets

in a p oset exist then every nonempty nite subset has a greatest lower b ound

Thus every meetsemilattice has a least element its greatest lower b ound and

every joinsemilattice has a greatest element

A nonempty intersectionclosed family of sets is a meetsemilattice

with the meet op eration given by intersection a nonempty unionclosed family

of sets is a joinsemilattice with the join op eration given by union

A lattice is a p oset which is b oth a meet and a joinsemilattice Lattices

can b e dened algebraically using the prop erties of the meet and join op erations

listed in Section If L is a meetsemilattice and U L has an upp er b ound

then U has a least upp er b ound U given by the greatest lower b ound of the

set of upp er b ounds of U This implies that if L has a greatest member then L

is a lattice If the semilattice L do es not have a greatest member it suces to

adjoin a new greatest element to L obtaining the lattice L L fg this is

the completion of L

The subset M of the meetsemilattice L is a meetsubsemilattice of L

i M and for every u v M u v is in M Joinsubsemilattices are dened

dually M is a sublattice of the lattice L i M is b oth a meetsubsemilattice and

a joinsubsemilattice of L

Let L and M b e meetsemilattices The map f L M is a meet

homomorphism i for every u v L f u v f u f v If f is a meet

homomorphism f is said to preserve meets Note that a meethomomorphism

is an orderpreserving map Joinhomomorphisms are dened dually If L and

M are lattices and f is b oth meet and joinpreserving then f is a lattice homo

morphism A semilattice isomorphism is a meetpreserving bijection A lattice

isomorphism is a meet and joinpreserving bijection

The element u of L is joinirreducible i whenever v w u either

u v or u w If u covers two distinct elements v and w then u v w

This implies that u is joinirreducible i u covers at most one element of L The

only joinirreducibl e of L which covers no other element of L is The proper

joinirreducibl es of L are the joinirreducibl es of L other than Note that every

element u of L is the join of the joinirreducibl es b elow u Meetirreducibles are

dened dually

The semilattice L is atomic i every prop er joinirreducible of L is an

atom of L The joinsemilattice L is coatomic i every prop er meetirreducible

of L is a coatom of L

The Boolean lattice B generated by n atoms is isomorphic to the

family of subsets of an nset The atoms of B corresp ond to the oneelement

subsets

The distributive laws are the identities

u v w u w v w u v w u w v w

A distributive lattice is a lattice satisfying the distributive laws Note that

the dual of a distributive lattice is distributive Since union distributes over

intersection and intersection distributes over union a family of sets which is b oth

union and intersectionclosed is a distributive lattice in particular families of

ideals and families of lters of p osets are distributive lattices The converse is

part of the Fundamental Theorem of Distributive Lattices see Gratzer for

a pro of

Theorem For every distributive lattice L with L there is a unique

up to isomorphism poset P such that L is isomorphic to the family of ideals of

The p oset P in the theorem is given by the set of prop er joinirreduc

ibles of L The isomorphism is obtained by assigning to each element u L

the ideal of all prop er joinirreducibl es b elow u Since the family of ideals of P

is the family of lters of P Theorems and imply that the class of

distributive lattices L with L is equivalent to the class of p osets of the

form

Let P b e a p oset Consider the family A of antichains of P The

corresp ondence b etween antichains and ideals discussed in Section shows

that A with the ideal order is a distributive lattice where A B is given by the

set of maximal elements of A B The corresp ondence b etween antichains and

lters shows that A with the lter order is a distributive lattice where A B is

given by the set of minimal elements of A B These two lattice structures on A

are in general not the same An imp ortant subfamily of the family of antichains

of P for which the two lattice structures do coincide is the family of Sp erner

antichains of P

Theorem Dilworth Let A and B be Sperner antichains of the poset

P Then for both the lter and the ideal order of antichains A B and A B

are Sperner antichains of P In either case A B is the set of minimal elements

of A B and A B is the set of maximal elements of A B

Let L b e a lattice The upper covering condition for L asserts that if

u covers v in L then for every w L u w covers v w or u w v w

The lattice L is upper semimodular i L satises the upp er covering condition

The lower covering condition for L asserts that if u covers v in L then for every

w L u w covers v w or u w v w The lattice L is lower semimodular

i L satises the lower covering condition The lattice L is modular i L is b oth

upp er and lower semimo dular

The lattice L is geometric i L is atomic and upp er semimo dular A

subset B of the atoms of the geometric lattice L is independent i for every

W W

prop er subset A of B A B Indep endent sets of atoms have many of the

prop erties of indep endent sets of vectors of vector spaces

Theorem If x is an element of the geometric lattice L then there is an

independent set of atoms A such that A x If x y and A is an independent

set of atoms such that A x then there is an independent set of atoms B

such that B A and B y If A is an independent set of atoms and a is an

atom with a A then A fag is independent

Two Results from the Combinatorics of Sets

Theorem Let F be a tree on an nset Let fU U g be the family of

members U of F with U Suppose that for i b U covers r sets

i i

in F Then

F n r

Pro of Let X b e the domain of F We can assume that U is maximal in

fU U g For i let F consist of the singletons of X and the sets U for

i i j

j i Let M b e the set of maximal members of F Since F is a tree for

i i

U V F U and V are either comparable or disjoint It follows that M is a

partition of X for each i If i j then U U so if U covers U in F then U

i j i

is maximal in F Thus U covers r members of M which gives

i i i i

M M r

i i i

for each i This implies that

M M b r

b i

Since M fX g and M ffxg j x X g this gives

r n b

Solving for b and using F n b yields

F n r

as required

Corollary If F is a tree on an nset then F n

U then U covers at least two members of F The Pro of If U F and

result follows by Theorem

The next result known as Kleitmans lemma has many applications

Its pro of can b e found in Anderson

Theorem Kleitman Let X be an nset If F and G are lters in

then

F G F G

n n n

This says that the density in of the intersection of two lters is at

least the pro duct of the densities of each

Notes

Section The terminology for induced families of sets and re strictions of families of sets given in Table lo osely follows that in Lovasz

Section Families of sets are a fundamental research area of

combinatorics An excellent text on the sub ject is Anderson

Families of sets are often studied as hypergraphs Hyp ergraphs are

socalled b ecause they are a natural generalization of graphs An uptodate

overview of research on hypergraphs can b e found in F uredi

The notion of a tree is usually dened for graphs In graph theory a

tree is a connected graph without circuits An arborescence is a directed graph

with ro ot a such that for every vertex x of the graph there is a unique path

from a to x Arb orescences can b e obtained from trees by selecting a ro ot and

directing every edge of the graph away from the ro ot A tree of sets F can b e

made into an arb orescence by connecting U to V i U covers V in F Conversely

an arb orescence in which every vertex has either outdegree zero the leaves or

outdegree at least two can b e made into a tree of sets by asso ciating to each

vertex v the set of leaves which can b e reached from v

Trees and arb orescences are well researched structures and have many

applications in combinatorics see Lovasz as well as in computer science

see Cormen et al and classication theory

Section Posets are used in all areas of mathematics The com

binatorial prop erties of p osets have attracted a lot of interest particularly since

Rotas article on the theory of Mobius functions Yet in available

texts which cover p osets the fo cus is usually on lattice theory Crawley and

Dilworth Gratzer For an account of the theory of p osets from a combi

natorial p ersp ective including a collection of interesting exercises with solutions

and references see Stanley Go o d sources of information on current research

in this area are Pro ceedings and

Altwegg shows that a p oset can b e dened in terms of its b etween

ness relation up to duality Using Hasse diagrams to represent p osets is stan

dard practice An interesting op en problem is to characterize the undirected

graphs which arise from Hasse diagrams of p osets see

The fact that every p oset has a linear extension is mentioned as the

rst of three fundamental results in the theory of ordered sets in Rival He

attributes this result to Szpilra jn

Every subset of a p oset can b e considered as a subp oset with the in

duced order However this notion of subp oset is to o general for many purp oses

A go o d alternative is the retract The subset Q of P with the induced partial

order is a retract of P i there is an orderpreserving map f from P to Q which

is the identity on Q The retract construction is discussed in Rival

Chains and antichains gure prominently in the theory of p osets The

height and width of a p oset are two of the most imp ortant invariants

Sp erner antichains are named after E Sp erner In he proved that

the width of the family of subsets of an nset is This result known as

b c

Sp erners theorem is the starting p oint for the study of the combinatorics of

sets Anderson

Dilworths chain decomp osition result is the second fundamental result

mentioned by Rival in the third is the xed p oint theorem for lattices due

to Knaster and Tarski

Posets and orderpreserving maps constitute the category of p osets

Exp onentiation Q of p osets has many of the prop erties of exp onentiation of

sets The op erations of exp onentiation cartesian pro duct and disjoint sum can

b e used to dene an arithmetic of p osets This is discussed in Jonsson See

also Birkho pp

Section Lattices are the b est studied class of p osets These

structures are studied from b oth an algebraic Crawley and Dilworth

Gratzer and a combinatorial Stanley p ersp ective The Fundamental

Theorem of Distributive Lattices is sometimes used to argue that the study of

p osets essentially reduces to the study of distributive lattices

The structure of the lattice of antichains has attracted substantial at

tention For a generalization of Theorem and some surprising duality

results see Greene and Kleitman and Greene

The set of bases of a geometric lattice is often called a matroid Ma

troids o ccur naturally not only as the sets of bases of vector spaces but also in

graph theory and the study of matchings in hypergraphs Matroids and geomet

ric lattices are discussed in Crap o Gratzer and Welsh

The expression for the size of trees in Theorem is usually

stated as an identity involving the number of leaves and the number of no des

for nary trees see Grimaldi

Kleitmans lemma is considerably strengthened by an inequality due to

Ahlswede and Daykin see Anderson

CHAPTER

FAMILIES OF SETS WITH WIDTH RESTRICTIONS

Denitions

Let F b e a family of subsets of the set X

Denition The center of A X is given by

CA fx A j if x U F then U A or A U g

Thus CA consists of the elements of A not contained in any member of F

incomparable to A The family F is centered i for every U F CU

Denition The family F is locally k wide i for every x X the width of

F is at most k Equivalently F is lo cally kwide i every antichain A F

fxg

with A k has empty intersection

Since X and the singletons of X are either comparable to or disjoint

from every subset of X

Observation If F is locally k wide then so is the family F F fX g

ffxg j x X g

Observation Cfxg fxg and CX X

Observation If F is centered then so is the family F F fX g

ffxg j x X g

Denition The family F is a pseudotree on X i F is centered X F and

for every x X fxg F The family F is a kpseudotree on X i F is a lo cally

kwide pseudotree

Since F is lo cally wide i for every U V F U and V are either

comparable or disjoint

Observation F is a pseudotree i F is a tree

The Structure of Centered Families of Sets

Let F b e a family of subsets of X Since the center of a set U do es not

intersect any member of F incomparable to U

Observation For every A X the family fU F j CU Ag is a chain

Observation Let A be an antichain of F with A Then the centers

of the members of A are pairwise disjoint and disjoint from A

It follows that if F is centered and A is an antichain of F with A

S T

k This gives the following observation A and A then

Observation If n and F is a centered family of subsets of an nset

then F is an n pseudotree

Theorem Let U V F If U V and U CV then CU CV

Pro of We show that U n CV U n CU Let x U n CV Then

x V n CV so there is a set W F such that x W and W is incomparable

to V The assumptions on U and W CV imply that W is incomparable

to U Hence x CU as required

Corollary The family fCU j U F g of centers of members of F is a

forest

Pro of Let U V F such that CU CV Then U and V are

comparable By Theorem CU and CV are comparable

Theorem If F is a centered family of sets and U F then

wF wF

Pro of Let w wF and w wF Since F F fU g

U U

CU CU

w w To show that w w let fU U g b e an antichain of F

Since U intersects CU the sets U and U are comparable for each i If U U

i i i

for each i then fU U g F which implies that w w Supp ose that

w U

for some i U U Then by Theorem CU CU Observation

i i

implies that w w

Theorem Let F be a centered family of arcs of Z which contains Z

n n

and the singletons of Z Then F is a pseudotree

Pro of Supp ose that F has three pairwise incomparable arcs U x x

l r

V y y and W z z such that U V W Let x U V W We

l r l r

can assume that in clo ckwise cyclic order

x y z x x y z

l l l r r r

This implies that V U W so the center of V is empty contradicting the

assumption that F is centered

Basic Extremal Results

Theorem If F is a centered family of subsets of an nset then F

This bound is best possible

Pro of To show that the b ound is attained

Example Let

F f i fj g j i n i j ng

so that F consists of the initial segments of n with a singleton adjoined Then F

is a centered family of subsets of n where the center of the member i fj g

of F is fj g The cardinality of F is

Pro of of the b ound Let r n and consider the family F of r

element members of F Let U F Since the members of F are pairwise

r r

incomparable if V F and V U then CV U This implies that

U F n fU g n hence

F n r

The b ound is obtained by summing this inequality for r n

Theorem If F is a centered family of segments of n then F n

This bound is best possible

Pro of To show that the b ound is attained

Example Let

F ffig j i ng f i j i ng fi n j i ng

so that F consists of the singletons of n and the segments of n containing

either or n In any family of segments of n C i and n Cj n

Thus F is centered The cardinality of F is n as desired

Pro of of the b ound By Observation we can assume that F contains

n and the singletons of n By cyclic closure families of segments of n can

b e considered as families of arcs By Theorem any centered family of

arcs of Z which contains Z and the singletons of Z is a pseudotree By

n n n

Theorem b elow n is the b ound on the size of pseudotrees on an

nset so the result follows

Theorem If F is a locally k wide family of subsets of an nset X and

r then F

r r

Pro of Every element of X is contained in at most k members of F Hence

k n k X F r

From this we can deduce a b ound on the maximum size of a lo cally

kwide family of sets

Theorem If F is a locally k wide family of subsets of an nset then

F n k n ln n

Pro of For n

F F F F

k n k n

n k n lnn

A b ound on the size of lo cally k wide families of sets linear in the size

of the domain will b e obtained in Section

The Size of k pseudotrees

Theorem Let n k If F is a k pseudotree on an nset X then

k k

F k n This bound is best possible

By Observation if n a pseudotree on an nset is an n pseudotree

Therefore the assumption that n k do es not restrict the generality of the

theorem

Pro of To show that the b ound is attained

Example Let

F ffig j i ng f i fj g j i j min i k ng

so that F consists of the singletons of n and the initial segments of n with

one of the next k singletons adjoined

k k

The cardinality of F is n n n k k n

To see that F is centered let U F If V F and V contains the

greatest integer in U then either V U or U V Therefore j CU

To show that F is lo cally k wide supp ose that U fU U g

is an antichain of F For each i let x b e the greatest integer in U Since

i i

x CU for each i the x are distinct Without loss of generality assume

i i i

that x x x Then x x k so if U fx g then

k k k k

U U contrary to assumption Therefore U fx g which implies

k k k

that U as desired Note that this argument shows that the family of

nonsingleton members of F has width k

Pro of of the b ound By asso ciating some of the sets in F with their centers

in a onetoone way and then asso ciating the remaining sets in F with elements

of X in a k toone way we will rst establish an upp er b ound of k n The

b ound will then b e reduced to k n by an analysis of these asso ciations

To further reduce this to the b ound given in the theorem a notion of deciency

will b e dened for arbitrary members of k pseudotrees The deciency of X will

b e the minimum value of k n F

Let C b e the set of centers of members of F By Corollary and

since F contains X and the singletons of X C is a tree By Observation

for every C C the family fU F j CU C g is a chain Asso ciate the

minimal member of this chain with C Since the maximum size of a tree on an

nset is n Theorem this asso ciation which is onetoone accounts

for C n members of F

Let F consist of the members of F not asso ciated with their center

For every U F we will select an element pU U n CU This will b e done

in such a way that if pU pV and U V then U and V are incomparable

Lemma Let U F Then there is an element x U n CU such that

for every W F with x W U x CW

Pro of Let V b e the maximal member of the chain of sets

fW F j W U and CW CU g

Let V b e a maximal member of

fW F j W U and W U n V g

Since the singletons of X are in F such sets exist and V is welldened If V

V then by Theorem CV CV CU contradicting maximality of

V Therefore V and V are incomparable and CV U n V Let V V

V b e a strictly decreasing chain of members of F such that

i V fxg for some x X

ii V is a maximal member of

fW F j W V and W CV g

j j

Such a chain exists To show that x is as desired let W b e a member of F with

x W U By Theorem the centers of the V form a decreasing chain

By the maximality condition on V W V Since W intersects the center of

V W is comparable to V for each j In particular W V Let j b e the

j j

greatest index such that W V Since V fxg if j r then W V If

j r j

j r then the maximality condition on V gives W V Hence x CW

j j

For each U F let pU b e an element of U n CU which satises

the conclusion of Lemma If pU pV and U V then U and V are

incomparable For every x X dene

sx fU F j pU xg

and

l x wF

fxg

F k n Since Since F is lo cally k wide sx l x k Therefore

F n F C

F C F

C sx

k n

Let X fx X j sx l xg

C n X Lemma

Pro of For x X let C b e the minimal member of

fC C j x C and for every U F with pU x CU C g

Since CX X such sets exist and C is welldened Let C fC j x X g

x x

For C C let r fx X j C C g We will show that if C C then C

C x

r covers at least r members of C Since X and by the b ound

C C

C n X on the size of trees in Theorem it will follow that

Let x X Let U fU U g b e the family of members U of

F with pU x Let C b e the center of U Then x C and since U is an

i i i

antichain the C are disjoint Observe that l x Pro of If l x then

the sets U F with x U form a chain This implies that if x U F then

x CU so that x pU Hence sx contradicting sx l x

Lemma If C C contains x and C C for some i then C C

i x

Pro of Without loss of generality assume that C intersects C Let W b e a

set in F with center C Since x C the U are comparable to W Since W

intersects C and by Theorem if W U then C C which contradicts

x C Therefore W U For i U and U are incomparable so W U

i i

for each i Since x U C we have C C for each i By minimality of C it

i i x

follows that C C as required

Lemma If C C and C C for each i then C C

i x

Pro of It suces to show that x C Let W b e a member of F with center

C Since the C are disjoint C C and therefore W U for each i Supp ose

i i i

that x C Let V b e a set in F incomparable to W such that x V Since C

and V are disjoint V and U are incomparable for each i It follows that fV g U

is an antichain with x fV g U contradicting l x U

Lemma Suppose that y X with y x and C C If C is a

y x

member of C with x C and y C then C C

Pro of Let V fV V g b e the family of members V of F such that

pV y Let W b e a member of F with center C

Supp ose that y U for some i Then W U Since U intersects C

i i i

we have U W and C C so that by Lemma C C By symmetry

i i x

if x V for some i then C C C

i y x

T T

Supp ose that y U and x V Since l y is the maximum size

of an antichain of F U is comparable to at least one of the V We can

fy g

assume that U is comparable to V If U V then by the prop erty expressed

in Lemma y C so that C intersects C and by Lemma C C

The case V U is symmetric This completes the pro of

Let C C and let fx x g b e the set of elements x X such

that C C so that r r Supp ose that C covers C in C Then C

x C

contains at most one of the x Lemma If x C then C do es not

i i

include the center of any set U with pU x Lemma Consider x

If U fU U g is the family of U F with pU x then there is an

i such that C do es not include CU Lemma Since CU C for

i i

each i and l x this implies that C covers at least r members of C as

required

Lemma yields

F C sx

n X sx

For x X sx l x and for x X n X sx l x This gives

F n l x

n l x

k l x k n

Let dx k l x Then

F k n dx

The number dx is called the deciency in F of x For U F the deciency

in F of U is dened by

dU dx

xCU

Let dm l b e the minimum value of the deciency in G of U for arbi

trary k pseudotrees G and members U of G such that l wG and m is the

cardinality of the center in G of U Dene

minmk l

k l i bm l

Lemma dm l bm l

k k

In particular this shows that dn Since dn is the deciency

F k n dn of X the center of which is X we obtain

k k

k n thus completing the pro of of the theorem

Pro of If l k then the lemma asserts that dm k k k

Since dx for each x the inequality dm k bm k holds

Supp ose that l k The remainder of the pro of pro ceeds by induction

on m Let U b e a set in the k pseudotree G with center C such that wG l

C m We compute l x sx dx and CV relative to G Let C b e and

the tree of centers of members of G

Let m Then C fxg for some element x and dU dx

k l x To show that dU k l observe that by Theorem l x l

Let m Since by Theorem the width of G dep ends only on

CV we can assume that U is minimal among the sets V of G with CV C

Let U consist of the sets V of G such that for some C C covered by

C V is the maximal member of the chain fW G j CW C g Since C

and the singletons are in C C covers at least two members of C Therefore

U If C covers C in C and C is the center of V G then V U This

implies that every member of U is included in U

We show that U has at least two maximal members Let U b e maximal

in U Let x C n CU Let V b e maximal among the members W of G such

that W is incomparable to U and x W Since V is disjoint from CU C

but intersects C it follows that V U and CV C Let C b e the member

of C which contains CV and is covered by C Let U b e the member of U with

center C Then U V so that U U Since U is maximal in U U and

U are incomparable By the maximality condition on V U V and U is

maximal in U as desired

Write U fU U U g where U is maximal in U fU U g for

t i i i t

each i and U and U are maximal in U Let c CU Let s c c

i i i i

Note that s m Let r b e the least index i such that s k or i t

t i

Lemma For i r wG maxs l For i

U i

wG l

Pro of Let w wG Let V fV V g b e an antichain of G

U w U

i i

Since V intersects C V is comparable to U for each j Supp ose that for some

j j

j V U Then V G and therefore w l Supp ose that V U for each

j U j

j Then CV C for each j Hence for every j there is a U such that

k j

V U and CV CU We have U V for each j so by maximality

j j i j

k j k j

of U and U in U if i or i then w Supp ose that i Since U

is maximal in U either w or the V are not included in any member of U

i j i

If w we are done If w then k j i for each j This implies that

i w w

CU CU CV

j j

k j

j j j

Since CU s and the centers of the V are disjoint w s as

j i j i

required

Since

dU dU dU

dc wG dc wG

r U U

and c m for each i we can apply the induction hypothesis to b ound dU

from b elow For i r let l maxs l For i and i let

i i

l Since the l l Let r min c k l By Lemma wG

i i i i i U

inductively obtained lower b ound bc l for dc l is decreasing in l it

i i

follows that

dU r bc l bc l bc l

r r

k l k l i

k l i k l

r r

Since the sum of the leading s is r

dU k l k l i

k l k l i

r r

By using l l l and reducing some of the terms we obtain

dU k l k l k l r

k l k l k l r

r r r r

Let q min r r k l and for i let q r We have

i i

k l r k l so that we can combine the rst two lines of the

ab ove inequality to obtain

k l k l k l q

r k l k l k l q

r r r r

Note that each line is a sum of successive nonnegative integers Let T b e the

j th term of line i and let T b e the sum of the T To complete the

pro of we show that T bm l

If q k l or q m then

minmk l

X X

k l j bm l T T

j j

Supp ose that neither q k l nor q m Then

i r c k l

ii r c k l

iii q c c s

iv s m

v s k l k

Inequalities iv and v imply that r Let r b e the least index i such that

q k l or i r By i and ii r Let i r We show that

i i

T T If i we have

iq i

T k l q

k l s

k maxs l

k l

Supp ose that i Since q r k l we have q c Using

i i i i i

l maxs l maxs l c l c we obtain

i i i i i i

T k l r

iq i i

k l c

i i

k l c

i i

k l

Observe that

q s

i r

We can now b ound Let q s q

r r

X X

T T

j i

X X

i j

minq k l

k l j

k l It remains to show that min q k l minm k l Either q

r r

l Then either l k l q c Supp ose that q or r r and q

r r

r r r r

k l and the result follows If l then q q s If l or l

r r r r

s then q s k s k and we are done Supp ose l

r r r r

c Then q s By denition of r s min m k that r r and q

r r r

This gives minq k l min m k l as desired

The pro of of the theorem is complete

Intersecting Antichains of Arcs

A k and Theorem If A is an antichain of arcs of Z such that

A then every A A has at least k elements

Pro of Supp ose that A fAg Then A implies that A contains at

least one element Supp ose that A Let A A and x A Then

A Z so that A a a for some a a Z By rotation we can assume

n l r l r n

that a x a If B A and B A then B has either a left endp oint

l r

b with a b x or a right endp oint b with x b a but not b oth Since

l l l r r r

A is an antichain no two members of A have the same left or the same right

endp oints Therefore

a x n fa g x a n fa g k

l l r r

A k which implies that

Corollary If F is a maximal locally k wide family of arcs then F con

tains the ielement arcs for i k

The Size of Lo cally k wide Families of Arcs and Segments

Theorem If n and F is a locally k wide family of arcs of Z then

F k n k

Let Ak n b e the maximum cardinality of a lo cally k wide family of

arcs of Z

The b ound on the size of lo cally k wide families of arcs is not optimal

Since for n k the family of arcs of Z is lo cally k wide we have Ak k

k k and Ak k k k For n k there are at most

k many k element arcs in F so that Ak k k k k This

shows that Ak n k n k k for n k k k It is conjectured

that Ak n k n k k for all n k

Example Let

F fi j j i j k or i k g fZ g

so that F consists of the ielement arcs for i k Z and the arcs with left

endp oint j where j k

By Theorem and its corollary to show that F is lo cally k wide

it suces to consider the family F of ielement arcs in F with i k

Since F is included in the disjoint union of the k chains fi j j j Z g for

i k wF k which implies that F is lo cally k wide as

k k

required

The number of ielement arcs of Z with i k is k n including the

empty arc There are n k many j element arcs with left endp oint i and

k j n Therefore

F k n k n k k n k k

Pro of of Theorem Let F b e a lo cally k wide family of arcs of Z

To prove the b ound we can assume that F contains Z and the singletons of

Z For k F n fg is a tree so by Theorem F n and the b ound

follows

Let k We assume that the b ound holds for lo cally k wide

families of sets We will reconstruct F from a k wide family of arcs and a

tree derived from F

In this pro of ordering relations involving more than two terms are

assumed to b e in clo ckwise cyclic order

Recall that if A a a and B b b are incomparable arcs and

l r l r

a B then the left overlap of A with B is the arc b a For every arc U F

r l r

let LU b e an arc in F which is incomparable to U and has maximal left overlap

with U If there is no such arc let LU Let U b e the left overlap of LU

with U Let F fU j U F g Let F fU n U j U F g Note that if U F is

U is nonempty nonempty then U n

Lemma F is a locally wide family of arcs

U intersects V n V We show Pro of Let U V F and supp ose that U n

that U n U and V n V are comparable Since Z n Z Z and we can

n n n

assume that U u u and V v v for some u u v v Z The right

l r l r l r l r n

U and V n V are u and v resp ectively By interchanging U endp oints of U n

r r

and V if necessary we can assume that u V n V If u v the U n U and

r r r

V are comparable and we are done Supp ose that u v If U and V are V n

r r

incomparable then U overlaps V from the left and the left overlap of U with V

strictly includes V contradicting the denition of V Therefore U V This

implies that LV overlaps U from the left Since the left overlap of LV with

U is included in U it follows that U n U V n V as desired

Lemma F is a locally k wide family of arcs

U fU U g b e an antichain of F with U Let x Pro of Let

U Then x U LU for each i Let U fU U g and LU

i i i i i

fLU LU g To show that l k we will construct an antichain

V U LU such that V l Since U LU F and F is lo cally

l l l l

k wide we can then deduce that l k as required

For i l let u u U and v v LU Then U

il ir i il ir i i

u v Since the U are pairwise incomparable u u and v v for

il ir i il j l ir j r

i j By reordering we can assume that

u u u x

l l ll

This implies that

x v v v

r r lr

We also have

v u x v u

il il ir ir

for each i

We collect some facts ab out the relationships b etween the U and the

i For i j LU overlaps U from the left since v u u x

i j il il j l

v v u

ir j r j r

ii For i j U U since u u x

i j il j l

iii For i j LU LU since x v v

i j ir j r

For i l we recursively construct antichains V U LU

i i i

such that V i and U V V need not include V

i i i i i

Let V fLU U g Assume that i and the antichain V has

b een constructed By fact ii either U and U are incomparable or U U

i i i i

We consider these cases in turn

Supp ose that U and U are incomparable Let V V fU g To

i i i i i

see that V is an antichain rst note that by fact i U is incomparable to each

i i

LU V To show that U is incomparable to each U V supp ose that

j i i j i

j i and U V Since U V U and U are incomparable Since

j i i i j i

u u u incomparability of U and U implies that x u u

j l il i i ir

il ir

and incomparability of U and U implies that x u u Therefore

j i j r

u u x u u

j l il j r ir

so that U is incomparable to U as required

i j

Supp ose that U U Then

i i

u u x v u u

il ir ir

il ir

Let V V n fU g and V V fU LU g To show that U and LU

i i i i i i i

are incomparable to the members of V we rst show that

u v u x

il il

To that end assume that u mo dn LU Then LU is incom

i i

parable to U and has left overlap u v with U Since U

i ir i i

u v u v this contradicts the maximality condition in the

il ir il

denition of U Hence u mo dn LU and follows

i i

Using and the other known relationships b etween the endp oints for

j i we obtain

v u u v u x v v u u

j l j l il il j r ir ir

il ir

This implies that U and LU are incomparable to LU

i i j

If U V then U and U are incomparable so that U overlaps

j j i j

U from the left By the maximality condition in the denition of U u

i i j r

v u Therefore using

ir ir

u u v u x u v v u

j l il il j r ir ir

il ir

which implies that U and LU are incomparable to U as required

i i j

This completes the construction of the antichains V

We construct a surjective map from the disjoint union of F and

F n fg onto F For A F let A b e the minimal member of the chain of

arcs U F with U n U A For A F n fg let A b e the minimal member

U A Note that the arcs of Z with left right of the chain of arcs with

endp oint i form a chain for each i Z

To show that is onto let U F If U U is the smallest arc

U U F whence U n U U Supp ose that U and with U n

U U Then U Z so U u u for some u u Z and there is an

n l r l r n

arc V v v F such that V U and V U We have V n V and

l r

U Let W w w b e an arc in F with W n W U n U Supp ose that U n

l r

W U and let x b e the left endp oint of U n U Then

u v w x v u w

l l l r r r

W v W However Thus V overlaps W from the left so that by denition of

V U implies that v U n U contradicting the assumption that U n U W n W

It follows that W U so that U is minimal among the arcs W F with

W n W U n U and thus U n U U as required

F n Since F Lemma implies Lemma implies that

that F n fg Ak n Therefore

F F F n fg

n Ak n

so that by arbitrariness of F

Ak n n Ak n

Iterating this recursive inequality and using the fact that A n n we get

Ak n k n k

as required

Theorem If n k and F is a locally k wide family of segments of n

then F k n k k This bound is best possible

This result follows from the next theorem which removes the restriction

on n Let Sk n b e the maximum cardinality of a lo cally k wide family of

segments of n

Theorem The function Sk n satises the recursive equations

S n

Sk for k

Sk n n Sk n for k and n

To see that this generalizes the b ound of k n k k for n k

asserted in Theorem let S k n k n k k We show that the

function S solves the recursive equations in the statement of Theorem for

k n Note that if k n then k n Compute S k n as

follows

S k n k n k k

n k n k k

n S k n

The only relevant b oundary condition is S n and since S n this

is satised

Pro of of Theorem The b ound is attained by the following example

Example Let

H fi j n j j i k g fi j n j i k g

k n

By cyclic closure H is equivalent to the family of arcs of Z in Example

k n n

excluding the arcs A with A Z A and n A In particular H is

n k n

lo cally k wide

To show that H S k n observe that H H

k n k n

H for k and H n Let k and n Let G b e the

k n

disjoint union of H and H n fg Dene H G by

n k n k n

i j H if i j i j or i

i j

i j H n fg otherwise

k n

Then is a bijection It follows that

H H H

k n n k n

n H

k n

H satises the recur This shows that the function S dened by S k n

k n

H S k n as required sive identities of the theorem so that k n

Pro of of the b ound Let F b e a maximal lo cally k wide family of segments

of n If k or n then F fg If n and k then F f fgg

so the b oundary conditions on the function S are satised If k and n

F n fg is a tree so that F n Since S n n S n n it

remains to consider the case k and n We assume that the b ound holds

for lo cally k wide families of segments

By considering F as a family of arcs on the cyclic closure of n we

can apply the reduction of the pro of of Theorem to F That is we obtain

a lo cally wide family F and a lo cally k wide family F such that there

is a surjective map from the disjoint union of F and F n fg to F The fact

F consist of segments that F consists of segments implies that b oth F and

Furthermore using notation from the pro of of Theorem if U is a segment

containing n no segment which overlaps U from the left contains n hence U

do es not contain n If U is a segment containing no segment overlaps U from

the left so U is empty This implies that F consists entirely of segments included

in n Therefore

F F F n fg

n S k n

By arbitrariness of F

S k n n S k n

as required

Maximal r antichains and Sp erner Closure

The results in this section are used in Section to obtain a b ound on

the size of lo cally k wide families of sets

Let P b e a p oset Let A b e the family of antichains of P We consider

A ordered by A B i A B the lter order Thus A is a distributive

lattice where A B consists of the minimal elements of A B If B A then

the antichain A B is maximal in B i for every B B A B Note that this

notion of maximality is dierent from the one induced by the inclusion order on

Denition The antichain A is a maximal r antichain of P i A is maximal

among the r element antichains of P A maximal antichain is a maximal

r antichain for some r Let C P b e the family of maximal antichains of P

If A is an antichain then A is the minimal antichain of the lter gen

erated by A Therefore

Observation A is a maximal r antichain i for every antichain B A

with B A B r

Supp ose that A and B are antichains and B A Then A n B B

is an antichain We have

A n B B A A B B

Thus A n B B A i A B B This implies

Observation A is a maximal antichain i for every antichain B A

A B B with B A

Theorem If A and B are maximal antichains then A B is a maximal

antichain

This theorem implies that C is a subsemilattice of A

Pro of Let A and B b e maximal antichains and dene

A A n B B B B n A A

A A B n B B B A n A

Then fA B A A A B B B g partitions A B and

A A B A A A

B A B B B B

A B A B A A B B

To show that A B is a maximal antichain supp ose that D is an

antichain of A B Let m A B A A D m A D

D A and n D n A Since m m is the number of elements of n

A b elow the antichain D A Observation implies that

m m n if D A A

m m n otherwise

Let m B B D A B D m Since B B

D contains the elements of B b elow the antichain D n A A D

Observation implies that

m n m if D n A A D B

m n m otherwise

If D A A or D n A A D B then we can combine the

inequalities ab ove to obtain

A B D m m n n D

Supp ose that D A A and D n A A D B Then D A B

Since B A n A D B Since A B D A Therefore

D A B A A B B A B

By arbitrariness of D and Observation A B is a maximal

antichain

If A and B are incomparable antichains then A B is strictly b elow

b oth A and B Therefore Observation implies

Corollary If A is a maximal r antichain and B is a maximal r antichain

where A and B are incomparable then A B is a maximal q antichain for some

q maxr r

Since the Sp erner antichains of P are the wP element antichains of

P the unique maximal Sp erner antichain of P is the least member of C

Corollary If A is the maximal Sperner antichain of P and B is a max

imal antichain of P then A B

Denition Let C b e the family of antichains A A such that if B is an

antichain of A and B A then B A i For i this agrees with

the denition of C If i the condition B A in the denition of C is

unnecessary Note that for i j C C

i j

The next observation generalizes Observation

Observation The antichain A is in C i for every antichain B A

A B i B with B A

The pro of of Theorem can b e adapted to show the following result

Theorem If A C B C and i j then A B C If i or

i j ij

j then A B C

maxij

This implies that C is also a meetsubsemilattice of A

Pro of Supp ose that i j If we follow the pro of of Theorem using

the same notation and applying Observation instead of Observation

then is replaced by

m m i n

and is replaced by

m j n m

The conclusion is

A B D m m i j n n D

which proves the result for i j

If i and j then A C so A B C C as required

maxij

Supp ose that i and j Then holds if D A A and

holds if D n A A D B Therefore as in the pro of of Theorem

A B D m m maxi j n n D

unless D A B as required

w w

This bound is best Theorem Let w wP Then C

possible

Pro of The b ound is attained by the following partial order

Example Let P b e the union of w disjoint antichains A A A

where A consists of w i elements Let x y i x A and y A for

i i j

w w

and for each i Ai is the only maximal w i i j Then P

antichain of P Figure shows P for w

d d

A A

d d d

Figure

Pro of of the b ound If A is a maximal r antichain of P then A is a maximal

r antichain of every subp oset of P which includes A Hence we can assume that

P P C and b ound

For every x P let C x fA C j x Ag By assumption

C x If A B C x then x A B and by Theorem A B C

which implies that A B C x Thus C x is closed under meets Let Ax b e

the minimal member of C x Then Ax is the maximum cardinality antichain

in C x

Lemma If x Ay then Ax Ay

Pro of Supp ose that x Ay Then x Ay Ax n Ay which

implies that Ay Ax Ay Minimality of Ax in C x and Ay Ax

C x imply that Ax Ay Ax Hence Ax Ay and the result follows

For r w let N r fx P j Ax r g The lemma implies

that for x y N r x y Hence N r is an antichain Let x N r By

the lemma if y N r then y Ax so that N r Ax Since Ax is a

maximal r antichain we have N r Ax r Using P N r we

obtain

P N N N w

w w

as required

Theorem Let A A A be a sequence of pairwise incomparable r

element antichains of a poset P of width w Suppose that for every i j A is

a maximal r antichain of A A with the induced order Then

i j

The b ound is attained by the sequence in any order of all r subsets

of the discrete w element p oset

Pro of Using Dilworths Theorem let P C C C b e a

decomp osition of P into disjoint chains For every antichain A of P let cA

cA A r It remains to show that for i j fC j C A g Then

i i

cA cA as this implies that l is at most the number of r element subsets

i j

of a w set

Supp ose that for some i j cA cA Then A A has width

i j i j

r so A and A are Sp erner antichains of A A By Theorem A A

i j i j i j

is a Sp erner antichain of A A Since A is a maximal r antichain of A A

i j j i j

A A A so that A A contradicting the incomparability assumption

i j j i j

Corollary If P is a poset of width w then P has at most maximal

r antichains

Let F b e a family of sets ordered by inclusion

Denition If A F and A is the maximal Sp erner antichain of F then

the Sperner closure in F of A is the set SC A A If F let

F A

SC A F Dene

SCF fSC A j A F g

SC F fSC A j wF r g

r F A

The uniqueness of maximal Sp erner antichains implies that SC is a

welldened op eration on the subsets of F

Observation For A F SC A A and SC SC A

F F F

SC A

Since F is a lter of F we have

Observation If A F and A is the maximal Sperner antichain of

F then A is a maximal antichain of F

Theorem For A B F SC A SC B

F F

Pro of Let A and B b e the maximal Sp erner antichains of F and F

A B

resp ectively Then B is a maximal antichain of F so by Corollary

T T

A B This gives SC A A B SC B

F F

Theorem SCF f A j A C F g

Pro of Observation implies that SCF f A j A C F g For the

reverse inclusion let A b e a maximal antichain of F and let B b e the maximal

Sp erner antichain of F Since A F A is a maximal antichain of

A A

F hence B A Corollary This yields

T T T T

A SC A B A

T T

so that A SC A SCF as desired

If A and B are antichains of F then

T T T

A B A B

Since C F is a meetsubsemilattice of the family of antichains of F The

orem implies that SC F is intersectionclosed This can also b e de

duced from the fact that SC is a closure op eration in the usual sense Obser

vation and Theorem

Corollary gives the following observation

Observation If A SC F and B SC F then A B SC F

i j k

where k maxi j The inequality is strict if A and B are incomparable

By Corollary if wF k then there are at most maxi

fxg

mal r antichains A F This yields

fxg

Observation If F is locally k wide then SC F is locally wide

If A B F and A and B are the maximal Sp erner antichains of

F and F resp ectively then as in the pro of of Theorem A B If

A B

A B then A B and A B This implies

Observation The families SC F are antichains of sets

The Size of Lo cally k wide Families of Sets

Theorem Let k and n If F is a locally k wide family of subsets

of an nset then F k n

The b ound k n is linear in n but sup erexp onential in k In

general unless n is much larger than k asymptotically as k unless

ln ln n k ln k the b ound of Theorem is smaller

For k the theorem implies that a lo cally wide family of sets F on

F n Since the maximum size of a lo cally wide family an nset satises

of arcs on Z is at least n Example this b ound is optimal up to a

constant see Corollary b elow

Whether the b ound on the size of lo cally k wide families of sets on

an nset is asymptotically linear in b oth k and n is an op en question The

largest known examples have k n members asymptotically for ok and

k on For comparison note that the asymptotic b ound for such families of

arcs is k n Theorem These examples are obtained by restricting lo cally

k wide families of subsets of Z to n They are constructed by enlarging the

families of arcs in Example

Example Let S consist of the k element segments of Z

S fi i k j i Zg

Let T consist of the segments of Z with left endp oint k

T fk l j l k g

For k let G b e given by

G f x i fx k g j x i x k x mo dk g

f fx k g i x j x i x k x mo dk g

Observe that if A B G then A and B are either comparable or disjoint

Dene F by

F fg fZg S S T T

F F S T G

k k k k k

Theorem The family of sets F is locally k wide for each k

Pro of The family F is an unbounded version of Example with k

Assume inductively that F is lo cally k wide Let x Z For each l let

H fA F j x A A l g

l l

Since F is lo cally k wide and by Dilworths Theorem H can b e decom

k k

p osed into k disjoint chains C C Since the maximal members of H are the

k k

k k element segments containing x the maximal member C of C is a k element

i i

segment for each i By denition of F H consists of the members of H

k k k

together with the k element segments from S and the chain C of

k k

members A of G with x A The members of C are included in a unique

k k

k element segment D of Z Since every k element segment is contained

in two successive k element segments we can match each C fC C g

i k

to a distinct k element segment D D with C D The chains C

i k i i

dened by C C fD g for i k form a decomp osition of H into

i i k

k disjoint chains

Let

H fA F j A k g

Then H consists of the k element segments and the union of the k

is lo cally disjoint chains fi j j j i k g for i k The fact that H

has k wide follows from Corollary This implies that H

fxg

a decomp osition into disjoint chains D D where the minimal member

of each chain D is a k element segment Thus the minimal members of

the chains D are the same as the maximal members of the chains C This

implies that the two chain decomp ositions can b e combined to yield a chain

decomp osition of F into k chains By arbitrariness of x F is

k k

fxg

lo cally k wide The induction is complete

To obtain a lo cally k wide family of sets on an nset we restrict F to

n let F F Then F is the family of segments of Example

k n k n

and has cardinality n for n

Let n k To estimate F observe that

k n

S n F fi i k j k i n k g n k

k k n

and

T n F S fk l j k l ng n k

k k n k

n n

The cardinality of R G n F S T can b e esti

k k n k k

n n n

mated as

n n

k k R

k k

n n

Since b ck n k and d ek n k

k k k k

this gives the following inequalities

F F F n k n k

k n k n k n

k k

Therefore for ok and k on F k n ok n

k n

For xed k these examples do not in general have maximum p ossible

size not even asymptotically in n as demonstrated by

Example Let

F fi j j j i g fi l j i g

f fi i g j i Zg

f fi i i g j i Zg

n while F n Then F is lo cally wide and F

Pro of of Theorem Let F fU U U U g b e a lo cally

k wide family of sets on an nset X Let F fU U g We can assume that

i i

U is maximal in F for each i Let

i i

G SC F SC F n fg

w i w w i

G Note that since for w k and i b Let G G and b

w w w b w

U SC F we have G F n fg for each i so that b b If i j

i i i i

then G G

w i w j

Lemma Let i b and B X If w F w then w G

i B w i B

Pro of Let fA A A g b e an antichain of G For j r

r w i B

Then A SC F and let f j b e the least j such that A is in G

j w j w j

f j

f j i for each j By reordering we can assume that f is nondecreasing Let

A b e the maximal w antichain of F with A A Since antichains with

j j j

f j

incomparable intersections are incomparable in the lter order the antichains

A are pairwise incomparable For j j F F F so that A is a

j i j

f j f j

A and A F It follows that the sequence maximal w antichain of A

j j i B

of antichains fA A g satises the conditions of Theorem whence

Lemma implies that G is lo cally wide In particular G is

w k

lo cally wide which gives b n An amortized counting argument will

for w w b e used to b ound the b in terms of the b

w w

Let w k write

G fA A A g

w b

Then A SC F and let f i b e the least index i such that A is in G

i w i

w i

f i

for each i By reordering we can assume that f is nondecreasing Let

H fA A A g

i i

Lemma If A is a nonempty member of SC F then A is maximal

w i

F for some in G Conversely if A is maximal in G then A is in SC

i w i w i

w w

Pro of Supp ose that A is a nonempty member of SC F By denition A

w i

is in G Let A b e a maximal w antichain of F such that A A If A is not

w i i

maximal in G then there is a member B of G with B A Consider such

w i w i

a B There is a w element antichain B of F such that B B Since B A

B A Since A is a maximal w antichain of F B A Theorem applied

to the sequence of antichains fB Ag yields w w F w However

i A

F with w w contradicting the fact A SC this implies that A SC

i w F

F are disjoint that SC F and SC

i w i

Conversely supp ose that A is maximal in G Let B SC A

w i F

F for some w w We show that Since w F w B is in SC

i i A

A B Since w F w F for each j it follows that for some

j B j B

B SC F G The inclusions j i w F w We have SC

w j w i j B F

A B SC B and maximality of A imply that A B

The assumption that f is nondecreasing and Lemma imply that

A is maximal in H for each i

i i

We consider a pro cedure which pro cesses each member A of G in

i w

turn At each step the pro cedure will p erform some token transactions on three

sets of tokens T T and T Initially

T X

o n

T G

w w

where denotes disjoint union The set of tokens T has copies of each

for k w w After the ith step T and T will satisfy set in G

T X n H

T fA H j A is maximal in H g

i i

The ith step will consist of adding A to T since A is maximal in H and

i i i either

i removing a total of at least two members from T T and T including

the members A of T with A A or

ii removing at least one member of T and no tokens otherwise

To see how this will b ound b let m b e the initial size of the set of

w w

with w w tokens T Note that m is determined by the b

w w

m b

k w w

The initial size of T is n No tokens are ever added to T or T The total

number of tokens added to T is b After a token has b een removed from T it

is never added again Since A is added to T in the last step the total number

of tokens removed from T is at most b or if b Let r b e the

w w

number of steps where only one token is removed and let r b e the number of

steps where two or more tokens are removed Then

b r r

When only one token is removed this token comes from T Therefore

r n

Since the total number of tokens removed is at least r r

r r n m b

w w

Combining these inequalities we obtain

b n b r n m b

w w w w

Solving for b

b n m

w w

We will estimate the righthand side of this inequality following the

description of the steps of the pro cedure

To describ e the circumstances under which the pro cedure removes to

kens from T we prove the following lemma

Lemma Let A be a maximal member of H such that A and A are

j i i j

where w incomparable and A A Then B A A is in G

i j i j

w F w

f i

Pro of To see that w w let A b e the maximal w antichain of F with

f i

T T

A A Let A b e a w element antichain of F with A A The

i i j j j

f j

inclusion F F implies that the width w of F is at least the

f j f i f i

width of A A Since A and A are incomparable the sequence of antichains

i j i j

fA A g satises the conditions of Theorem whence w w

j i

Supp ose that A is maximal in G Then by Lemma A

j j

w f i

SC F Since SCF is intersectionclosed and A SCF B A

i i

f i f i f i

as required F G A SCF By denition of w B SC

w w

f i f i

Supp ose that A is not maximal in G Since A is maximal in

j j

w f i

G we have f j f i so that A G Since f is nondecreasing

w f j w f i

G H hence A is maximal in G and by Lemma A

i j j

w f i w f i

SC F Let w b e the width of F Let

f i f i

A A B SC

i j F

f i

A A B SC

i j F

f i

A A Since Sp erner closure preserves inclu Since A SCF SC

i i i F

f i

sion

B A

Similarly

B A

The family F is obtained from F by the addition of the set

f i f i

U This implies that either w w or w w Let B b e the maximal

f i

w antichain of F with B B Let B b e the maximal w antichain

f i

of F with B B

f i

If w w then B B fU g so that B B This together

f i

with B A and A A B A implies that B B A A Therefore

j i j i i j

F G B SC

w w

f i

If w w then B B since B is the maximal Sp erner antichain of

F so that B B This together with B A and A A B

B i i j

f i

G F G A implies that B B A A Therefore B SC

j i j

w w w

f i

This completes the pro of of the lemma

Let

A fA H j A A and A is maximal in H g

i i i i

M fB j B is maximal in fA A j A H A A n A gg

i i i i i

A then A and A are Note that if A is maximal in H and A A n

i i i i

incomparable If B M then there is a maximal member A of H with

i i

for some w w A A B so that by Lemma B G

We now p erform the token removals required in the ith step of the

pro cedure Assume inductively that after the i th step and are

satised and that any tokens removed from T in the j th step for j i are

where w w F Note copies of B M taken from a copy of G

B j w

f i

that these conditions are satised initially after the th step Let C b e the

family of members of T included in A The pro cedure removes the members

of C from T and adds A to T The pro cedure also removes the elements of

S S

A n H from T If A H then ii is satised and the ith step is

i i i i

complete If A H but C then i is satised and the ith step is

i i

S S

complete Supp ose that A H and C Since A C we have

i i i

A whence M If C then A so that M by A n

i i i i i

maximality of A H It follows that to satisfy i it suces to remove the

i i

appropriate copy of each member of M from T

Let B M Let w w F Initially T contains

i B

f i

To show that there are such tokens left after tokens which are copies of B G

removed from T the i th step let t b e the number of copies of B G

was removed at the g j th step We can so far where the j th copy of B G

assume that g is increasing Let g t i Let A b e the maximal member of

H with A A B using the third inductive assumption We show

g g g

that fA A g is an antichain of G Let j j i Then

g g t w f i

A and A prop erly include B Consider the set A A We have

g j g j g j g j

A A B Let A b e a maximal member of H with A A

g j g j g j g j

Since B M and by construction of M A A B This implies

g j g j g j

that A A B so that A and A are incomparable as desired By

g j g j g j g j

w w

Lemma w G which implies that t as required

w f i

w w

To obtain the b ound b k n we show by induction on i that for

i k b k n Using the fact that b n observed earlier

k i

m n k n n

Using b n m

k k

b n k n n k n

Assume that b k n for j i Then

k j

k k j

m b b

k i k k j

k i k i

k k j

n n k n

k i k i

ij i

k k

n n k n

i i j

n k n

i j

The sum in this expression can b e approximated using the fact that j

m k n n

k i

i j

n k n

For i the expression in parenthesis evaluates to

i i i

Therefore

b n m k n

k i k i

This completes the induction step

We now have

b b k n

as desired

The estimated b ound k n obtained in the pro of of Theorem

ab ove can b e decreased to yield a slight improvement on the b ound for lo cally

k wide families of arcs in Theorem

Corollary For n the maximum size of a locally wide family of sets

on an nset is either n or n

The maximum size of an arbitrary family of sets on a set is

Pro of There are examples of such families of size n

Using notation from the pro of of Theorem we rst show that

G n By construction G Let A G Then there are incomparable

sets U and V in F such that U V A Let X F If X W X n A

then fU V W g is an antichain with nonempty intersection Since F is lo cally

wide W F This implies that wF so that no B G includes

X nA

X n A It follows that X G and either G do es not contain every singleton of

X or G has at least three maximal members Since G fX g ffxg j x X g

G n is a tree Theorem implies that

By the pro of of Theorem

b b n m

n n

n as desired

Notes

Section The history of the notion of centered families of sets

and k pseudotrees is discussed in the introduction

Section The result of this section app ears in Knill et al

The idea for the pro of of the b ound k n is due to Ehrenfeucht

Section The pro of of the linear b ound for lo cally k wide fam

ilies of arcs segments is based on a generalization of a pro of by Ehrenfeucht

which shows that pseudotrees of segments are linearly b ounded

Section Results on the structure of maximal r antichains were

discovered while establishing the linear b ound for lo cally k wide families of sets

The counting techniques used in the pro of of Theorem are similar to the ones used by Greene and Kleitman in

CHAPTER

POSET DENSITIES IN SEMILATTICES

Introduction

Let L b e a semilattice and P a p oset Recall that L denotes the

p oset of orderpreserving maps from P into L where f g i for every x P

f x g x

Denition Let F b e a lter of P and a a joinirreducibl e of L Dene

P P

TL F a ff L j f x a i x F g

An orderpreserving map f P L is of type F a i f TL F a

Observation For xed a L the families of maps TL F a partition

L into convex subfamilies one for each lter F of P

We are interested in comparing the cardinalities of the types One way

to do this is by the construction of matchings

Denition Let Q and Q b e subsets of a p oset A decreasing increasing

matching from Q to Q is a onetoone map Q Q such that for every

x Q x x x x

Denition The pair L a has the ful l downward matching property for P

L a MPf P for short i a is a joinirreducible of L such that for every

pair of lters F and G of P with F G there is a decreasing matching

P P

TL F a TL G a

Denition The pair L a has the top downward matching property for P

L a MPtP for short i a is a joinirreducible of L such that for every

P P

lter F of P there is a decreasing matching TL P a TL F a

Denition The pair L a has the weak downward matching property for P

L a MPwP for short i a is a joinirreducible of L such that for every

P P

lter F of P TL P a TL F a

Denition The semilattice L has the ful l top weak matching property for

P L MPf P L MPtP L MPwP for short i there exists a join

irreducible a L such that L a MPf P L a MPtP L a

MPw P

Observation If L a MPf P then L a MPtP If L a

MPtP then L a MPwP

For F P TL F This implies

Observation The pair L does not have any of the downward match

ing properties

It follows that the oneelement lattice has none of the downward matching

prop erties

Consider the oneelement p oset We have L L so that

TL a a and TL a L n a

Observation L a MPf i L a MPt and L a

MPw i a L

A fundamental problem is to determine the nontrivial semilattices

which have the full top weak matching prop erty for P This problem app ears

to b e very dicult even for the oneelement p oset The following infamous

conjecture is attributed to P Frankl

L then L MPw ie Conjecture If L is a semilattice with

there is a joinirreducible a L such that a L

This conjecture is usually stated in terms of unionclosed families of sets

see Section Although this problem is well known little progress has b een

made toward its solution See Notes at the end of this chapter for a discussion

of the history and available literature

The upward matching prop erties are dened analogously to the down

ward ones In order for these prop erties to b e nontrivial we have to explicitly

consider only the prop er joinirreducibles since otherwise L has every up

ward matching prop erty

Denition The pair L a has the ful l upward matching property for P

L a MPf P for short i a is a prop er joinirreducible of L such that for

every pair of lters F and G of P with F G there is an increasing matching

P P

TL F a TL G a

Denition The pair L a has the top upward matching property for P

L a MPt P for short i a is a prop er joinirreducible of L such that for

P P

every lter F of P there is an increasing matching TL a TL F a

Denition The pair L a has the weak upward matching property for P

L a MPw P for short i a is a prop er joinirreducible of L such that for

P P

every lter F of P TL a TL F a

Denition The semilattice L has the ful l top weak upward matching prop

u u

erty for P L MPf P L MPt P L MPw P for short i there

exists a prop er joinirreducible a L such that L a MPf P L a

u u

MPt P L a MPw P

Matching prop erties are downward by default Except for the next

section we will fo cus on the downward matching prop erties

Examples and Counterexamples

Let P b e a p oset

Example Consider the oneelement lattice the trivial lattice The

only orderpreserving map from P to is of type P Therefore has

none of the downward matching prop erties Since there are no prop er join

irreducibles has none of the upward matching prop erties

Example Consider the twoelement lattice For every lter F of P

T F consists of exactly one map f dened by f x if x F

F F

and f x otherwise Since f f i F G has all the matching

F F G

prop erties

Theorem If L a has one of the matching properties for P and M is a

semilattice then L M ha i has that matching property for P

Pro of Observe that for every joinirreducible a L

P P P

TL M F ha i TL F a M

P P

If TL F a TL G a is a decreasing increasing matching then the

map dened by

f hu v i h f u f v i

is a decreasing increasing matching from TL M F ha i to TL

G ha i M

Theorem will b e generalized in Section

Example Let n The Bo olean lattice B generated by n atoms is

isomorphic to the nfold cartesian pro duct of the twoelement lattice Ex

ample and Theorem imply that B a MPf P and B a

n n

MPf P for every atom a B It follows that for every lter F of P

P P P P n

TB TB B Hence by induction B F a P a

n n n

The following theorem together with the Fundamental Theorem of Dis

tributive Lattices Theorem shows that every nontrivial distributive lat tice has the full downward and upward matching prop erties

Theorem Let Q be a poset Let L be the family of ideals of Q If x is

maximal in Q then L x MPf P If y is minimal in Q then L y

MPf P

Pro of If z Q and I is an ideal of Q with I z then z I It follows

that z is joinirreducibl e in L for every z Q Let x b e a maximal element

of Q Supp ose that F and G are lters of P such that F G Dene

P P

TL F x TL G x by

f u n fxg if u F n G

f u

f u otherwise

Since x is maximal for every ideal I of Q I n fxg is an ideal Thus is a

P P

decreasing matching from TL F x to TL G x

Similarly if y is a minimal element of Q then we obtain an increasing

P P

matching TL G y TL F y by dening

f u fy g if u F n G

f u

f u otherwise

Note that by minimality of y if I is an ideal of Q then so is I fy g The

theorem follows

P P

Many atomic semilattices L satisfy TL P a TL a for

each atom a which implies that L MPw P This is illustrated by the

following example

Example Let M b e the semilattice consisting of and n atoms see

Figure The only prop er joinirreducible s are the atoms and for each atom

P P

a TM P a For n TM a so that M MPw P

n n

d d d d d d d d

Q Q

A A

Q Q

A A

Q Q

A A

Q Q

A A

Q Q

d d

M M

Figure

d d

Let M b e M with a greatest element adjoined Then M is a

n n n

P P

d d

TM TM for mo dular lattice For n a P a

n n

d d

every atom a M so that M MPw P

n n

A weak version of the full downward matching prop erty requires only

that for some joinirreducible a L the function f N dened by

f F TL F a is orderreversing ie F G implies that f F f G

d d

The lattice M is an example for which there are p osets P such that M do es

n n

not satisfy this weak version of the full downward matching prop erty for P

Example Let P b e the dual of M Then P consists of a k element

antichain A covered by The number of lters of P is Let a b e an atom

d d

of M The complement of a in M is isomorphic to M Thus

n n n

P P k

TM a M n n

Let f TM fg a Either f a in which case for every x f x

or f M in which case the restriction of f to P n fg can b e an arbitrary

TM fg a n map into the complement of a This implies that

k k

For k and n suciently large n n

Whether all nontrivial semilattices have the weak downward match

ing prop erty for every p oset remains an op en question There are lattices which

do not have the top matching prop erty for any p oset

Example Let L b e the unionclosed family of sets generated by the un

ordered pairs of adjacent vertices of the p entagon and the empty set Since L

has a least member L is a lattice with the join op eration given by union L is

atomic where the atoms are the edges of the p entagon

Let fa bg b e an edge of the p entagon Let fc ag and fb dg b e the

edges other than fa bg incident on a or b Consider the ideal I of L generated

by fc a b dg see Figure Then I fa bg and I n fa bg

This implies that there is no decreasing matching from fa bg to L n fa bg so

that L MPt

More generally let P b e an arbitrary p oset The sublattice I fa bg

of L is isomorphic to the fourelement Bo olean lattice B The subsemilattice

I n fa bg of L is isomorphic to M Example Since M is isomorphic

P P

to a prop er subsemilattice of B TI fa bg TI P fa bg Since

P P P

I is an ideal of L there is no decreasing matching TL P fa bg

TL fa bg

In Section we will show that every lattice which is isomorphic to a unionclosed family of sets generated by sets of size at most two has the weak

fc a b dg

fc a bg fa b dg

Z d d

Z Z

c t Z d d d td

fc ag fb dg

fa bg

B d t t

a b

Figure

matching prop erty for This implies that the lattice L in Example

satises L MPw

Semilattices Intersection and Unionclosed Families of Sets

For many purp oses the concepts of semilattices and intersection and

unionclosed families of sets are interchangeable Every semilattice has a canon

ical representation as an intersectionclosed family of sets where the meet op era

tion is intersection An equivalent representation as a unionclosed family of sets

where meet is represented by union is obtained by complementing every mem

b er of an intersectionclosed representation If L is a lattice then the canonical

intersectionclosed representation of the dual of L is the canonical unionclosed

representation of L where join is represented by union

Let L b e a semilattice

Denition Let P L denote the family of principal ideals of L Let J L

denote the set of prop er joinirreducibl es of L For every member u of L let

J u fa J L j a ug Note that J u J u J L u

If u and v are principal ideals of L then u v u v Therefore

Observation The family P L is intersectionclosed The map L

P L dened by u u is a semilattice isomorphism

Denition The canonical intersectionclosed representation of L is given by

F L P L

J L

the restriction of P L to J L

Theorem The family F L is an intersectionclosed family of sets with

domain J L The map J L F L is a semilattice isomorphism with inverse

given by J U U for U F L

Pro of Since F L is the restriction of P L to J L F L is intersection

closed Since J is the comp osition of the isomorphism from L to P L with the

restriction map u u J L it follows that J is a meethomomorphism

Every member u of L is the least upp er b ound of the prop er joinirreducibl es

b elow u ie u J u This implies that J is a semilattice isomorphism

Theorem The intersectionclosed family F F L has the fol lowing

property If a F and U is the minimal member of F with a U then

U n fag F

Pro of Let a F J L The minimal member of F is J a

fag

a J L Since a is a prop er joinirreducible of L there is a unique element u

of L covered by a Every member of L strictly b elow a is b elow u Therefore

J u J a n fag

Denition The family of sets F is an intersectionclosed representation of

L i F is intersectionclosed and F and L are isomorphic semilattices The

representation F of L is irredundant i no restriction of F to a prop er subset of

the domain is isomorphic to F

Theorem shows that F L is an irredundant representation of

L Every intersectionclosed family of sets has an irredundant representation

obtained by restriction The following theorem and its pro of show how to nd

such representations

Theorem Let L be an intersectionclosed family of sets Then there is a

onetoone map p J L L such that for each a J L pa a and the

map U fpa j a U g is an isomorphism from F L onto the restriction of L

to the image of p

Pro of For each prop er joinirreducible a L let ca b e the unique member

of L covered by a and let pa b e an arbitrary element of a n ca To show that

p is onetoone assume that pa pb Then pa a b Since pa ca

a b a Similarly a b b hence a b

Let X b e the image of p For each a J L and u L if pa u

then a u otherwise pa u a a This implies that for each u L

u X fpa j a J ug The result follows

Note that all irredundant intersectionclosed representations of L have

the same number of elements in their domains the number of prop er join

irreducibles of L In fact if F and G are irredundant intersectionclosed rep

S S

resentations of L then there is a bijection F G such that U F i

U G such bijections are called setsystem isomorphisms

The corresp ondence b etween semilattices and intersectionclosed fam

ilies of sets allows us to translate prop erties of semilattices to prop erties of

intersectionclosed families where the concept of a prop er joinirreducible is re

placed by that of an element in the domain In general this corresp ondence will

not b e mentioned explicitly and each member u of a semilattice L will b e iden

tied with the set of prop er joinirreducibles b elow u Conversely each element

x in the domain of an intersectionclosed family F will often b e identied with

the minimal member of F

fxg

Denition Let L b e a joinsemilattice Let M L denote the set of meet

irreducibles of L The canonical unionclosed representation of L is the family

F L given by

F L fM L n u j u Lg

Since M L J L F L fM L n U j U F L g where L is

the dual of L In F L the join op eration of L is represented by union The

results and denitions for canonical intersectionclosed representations of meet

semilattices translate to canonical unionclosed representations of joinsemilat

tices

Every semilattice L can b e made into a lattice the completion of L

by adjoining a greatest element if necessary Let L b e a lattice Then L is a

joinsemilattice

Observation The joinirreducibles of L are the generators of the canon

ical unionclosed representation of L

In Sections and we will use the family of generators of F L to dene a

notion of lo cality for L In those sections we will identify each u L with the

corresp onding member M L n u of F L

The canonical intersection and unionclosed representations of semi

lattices and their duals can b e used to obtain the following equivalent formula

tions of Conjecture

F Conjecture If F is an intersectionclosed family of sets with

then there is an element x F such that F F

fxg

Conjecture The unionclosed sets conjecture If F is a unionclosed

F then there is an element x F such that F family of sets with

fxg

Conjecture If F is a unionclosed family of sets with F and F

F F then there is a generator U of F such that

Conjectures and are equivalent by complementation The irredundant

intersectionclosed representations of semilattices show that is equivalent

to To see that is equivalent to use Observation and note

that if holds for lattices then it holds for semilattices Pro of Assume that

holds for lattices Let L b e a semilattice without a greatest element and let

L L f g b e the completion of L By assumption there is a joinirreducibl e

b b

a L such that L a Then

L a a L

as required

Sub direct Pro ducts of Semilattices

Denition The semilattice L is a subdirect product of the semilattices

L L i

i L is isomorphic to a subsemilattice of the cartesian pro duct L

ii for each i and u L there is a member hv v i of L such that

i n

v u

Condition ii implies that the restrictions y L of the pro jections are onto L

i i

If for some i y L is onetoone then L is isomorphic to L In this case L is

i i

a trivial sub direct pro duct of L L

Since the family of subsets of an nset is isomorphic to the nfold carte

sian pro duct of every intersectionclosed family F with F is a sub direct

F copies of Since every semilattice has an intersectionclosed pro duct of

L is a nontrivial representation it follows that every semilattice L with

sub direct pro duct The following example shows how to express a semilattice as

a sub direct pro duct with two comp onents

Let L b e a semilattice

Example Let A and A b e subsets of L such that A A J L Let

L P L and L P L For u L let

A A

u hu A u A i L L

Since A A J L is a onetoone meethomomorphism into L L see

Theorem It follows that L is a sub direct pro duct of L and L

Corollary b elow shows that every sub direct pro duct arises in this fashion

Example Let P b e a p oset Then L is a sub direct pro duct of P copies

of L Let x b e a nonmaximal member of P and P P n fxg Then every order

preserving map f P L can b e extended to an orderpreserving map on P

by letting

f x ff y j y xg

If L is a lattice this also works for maximal x P This implies that L is a

sub direct pro duct of L and L

The sub direct pro ducts of Example are full sub direct pro ducts

Denition The semilattice L is a ful l subdirect product of the semilattices

L L i L is a sub direct pro duct of L L and

n n

iii for every u v L if w u v then w u v

L L

The family Sub L of subsemilattices of L is intersectionclosed Thus

Sub L is a semilattice with meets given by intersection An alternate semilattice

structure on SubL is dened as follows

Denition Let SubL denote the family of subsemilattices of L For L L

Sub L dene L L by

L L fu v j u L v L g

Let L and L b e semilattices We can dene sub direct pro ducts op er

ationally

Denition The map f L Sub L is meetconcave i for every u v L

f u f v f u v If f L SubL is meetconcave then dene

L L fhu v i L L j v f ug

Theorem If f L SubL is meetconcave then L L is a

subsemilattice of L L

Pro of If hu v i and hu v i are in L L then by meetconcavity of f

v v f u u so that hu u v v i L L as desired

Theorem The semilattice L is a subdirect product of L and L i there

f u and exists a meetconcave map f L Sub L such that L

L L L

Pro of Supp ose that L is a sub direct pro duct of L and L For u L let

f u fv L j hu v i Lg

Since hu v i hu v i hu v v i f u is a subsemilattice of L Since hu v i

hu v i hu u v v i we have f u f u f u u This implies

that f is a meetconcave map whence L is isomorphic to L L Since

y L is onto L f u L

f u Let f L Sub L b e a meetconcave map with

L By Theorem L L is a subsemilattice of L L Since every

subsemilattice of L is nonempty y L L is onto L Since the union of

the range of f is L y L L is onto L It follows that L L is a

f f

sub direct pro duct of L and L

Denition Let L b e a sub direct pro duct of L and L Let L Sub L

and L Sub L b e the maps dened by

u fv L j hu v i Lg

v fu L j hu v i Lg

For i let w b e the minimal member of w The maps and

implicitl y dep end on L L and L By the pro of of Theorem the maps

are meetconcave L L and L L L L

Let u v in L Since is a meetconcave map u v u

i i i

i i

Hence u u v which yields u v

i i i i i

Observation The maps L L and L L are order

preserving

Theorem Let L be a subdirect product of L and L If hu v i is join

irreducible in L then either u is joinirreducible in L and v u or v is

v If both v u and u v then u joinirreducible in L and u

and v are both joinirreducible

Pro of Let hu v i b e a joinirreducible of L Since

hu v i hu ui h v v i

u v or v u Supp ose that u v To show that u is either

u u joinirreducibl e in L let u u u for some u u L Since

and u u

ui hu u i hu v i hu u i hu

Since hu v i is joinirreducible this implies that u u or u u as desired

v u v is joinirreducible in L Similarly if

The converse of Theorem holds if L is a full sub direct pro duct of

L and L

Theorem Let L be a ful l subdirect product of L and L Then hu v i is

joinirreducible in L i either u is joinirreducible in L and v u or v is

joinirreducible in L and u v

Pro of The if direction is given by Theorem

u Let hu v i L Assume that u is joinirreducibl e and v

hu v i L such that

hu v i hu v i hu v i

Since L is a full sub direct pro duct

hu v i hu v i hu u v v i

Hence u u u and v v v Since u is joinirreducible either u u

or u u Without loss of generality assume that u u Since v u is

minimal in u and v v it follows that v v so that hu v i hu v i as

desired

v is symmetric The case where v is joinirreducible and u

Theorem and inductive use of the representation of L as a sub

P nfxg

direct pro duct of L and L for nonmaximal x P yields the fact that

the joinirreducibles of L are the maps f P L such that for some join

irreducible a L and element y P f z a for z y and f z

otherwise

A sub direct pro duct L of L and L can b e viewed as an internal

sub direct pro duct where L and L are subsets of L closed under least upp er

b ounds

Denition Let L L Then L is a lubsubsemilattice of L i for every

W W

U L such that U exists in L U L ie L is closed under l east upp er

ounds of L b

If L is a lubsubsemilattice of L then L and for every

u v L

v fw L j w u v g u

so that L is a meetsemilattice the meet of L need not agree with the meet of

Observation The subset L of L is a lubsubsemilattice of L i L

v w L and for every u v L if u v w L then u

Theorem Let L be a subdirect product of L and L Dene L L and

L L by

L fhu ui j u L g

v v i j v L g L fh

Then for i L is a lubsubsemilattice of L and y L L L is a

i i

i i i

meetisomorphism

Pro of By symmetry it suces to prove the result for L Observe rst that

h i h i L

L L L L L L

i L so that h

L L L L L

u i hu u i exists in L Let u u L and supp ose that hu

Then u u exists in L and since u u u and u u u

Observation we have

u i hu u i hu u u u i hu

Thus L is closed under pairwise greatest lower b ounds of L It follows that L

is a lubsubsemilattice of L Since

hu u i u i hu u u u i hu

y L is a meetisomorphism onto L as required

Let L b e a sub direct pro duct of L and L and let L and L b e the

lubsubsemilattice s of L dened in Theorem The isomorphisms L L

and L L induce pro jections and of L onto L and L resp ectively For

i these pro jections satisfy

u fw L j w ug L

i L i i

and

u u u

for u L

Denition Let A L The lubsubsemilattice of L generated by A is given

B j B Ag G A f

The projection of L onto G A is dened by

A L

u u A

for u L This gives for i

Theorem The map is a meethomomorphism of L onto G A

A L

Pro of Let L G A The prop er joinirreducibles of L are included in A

This implies that the family P L is an intersectionclosed representation of

L The map L P L dened by u u A is a meetisomorphism

with inverse given by B B for B P L Let L b e

the meethomomorphism dened by u u A Then and the

result follows

Denition The semilattice L is the internal subdirect product of L and L

i L and L are lubsubsemilattices of L such that L L J L

Theorem If L is the internal subdirect product of L and L then L is

a subdirect product of L and L

Pro of Let L L L b e dened by

ui u u h

L L

By Theorem is a meethomomorphism from L into L L Dene

L L L by

hu v i u v if u v exists in L

Then

u u L u L

Since J L L L u u Therefore is onetoone whence L is

isomorphic to a meetsubsemilattice of L L Since and are onto L

L L

and L resp ectively L is a sub direct pro duct of L and L

Theorems and yield

Corollary The semilattice L is a subdirect product of L and L i L

L L and L is the internal subdirect product of L

Corollary The semilattice L is a subdirect product of L and L i there

are subsets A and A of L such that A A J L and P L L for

A i

Constructions which Preserve Matching Prop erties

Let L b e a semilattice Let P and Q b e p osets Let F and F b e lters

of P such that F F and let G and G b e lters of Q such that G G

Theorem If there are decreasing matchings

P P Q Q

TL F a TL F a and TL G a TL G a

then there is a decreasing matching

P Q P Q

TL F G a TL F G a

Pro of The matching is dened by

f y P x if x P

f x

f y Qx if x Q

P Q

for f TL F G a where f y P and f y Q are the restrictions of f to

P and Q resp ectively

The following is a partial converse of Theorem

Theorem Suppose that G G or that a is an atom of L If there is a

decreasing matching

P Q P Q

TL F G a TL F G a

then there is a decreasing matching

P P

TL F a TL F a

P Q P Q

Pro of For f L and g L let f g L b e the map dened by

f x if x P

f g x

g x if x Q

Let h TL G a b e the map dened by

a if x G

otherwise

Then h is the minimal member of TL G a

P Q P Q

Given the decreasing matching TL F G a TL

F G a dene by

f f h y P

P P

for f TL F a The denition implies that f TL F a and

f f for f TL F a It remains to show that is onetoone If

G G the only map h TL G a with h h is h h If a is an atom

the only map h TL G a with h h is the map h dened by h x a

if x G and h x otherwise This implies that f h y Q h for every

f TL F a Since is onetoone is onetoone

Theorems and have the following corollaries

Corollary L a MPf P Q i L a MPf P and L a MPf Q

Corollary If L a MPtP and L a MPtQ then L a

MPtP Q

Corollary Let a be an atom of L Then L a MPtP Q i L a

MPtP and L a MPtQ

Theorem Let I be an ideal of L If there is a decreasing matching

P P

TL F a TL F a

then there is a decreasing matching

P P P

y I TI F a TI F a

P P

Pro of Let TL F a TL F a b e a decreasing matching The

P P P

decreasing matching y I TI F a TI F a is obtained by restrict

ing the domain of to TI F a Since I is an ideal and is decreasing the

range of the restriction is included in TI F a

Corollary Let I be an ideal of L If L a MPf P then I a

MPf P If L a MPtP then I a MPtP

Let L and L b e semilattices The following theorem generalizes part

of Theorem

Theorem Let f L SubL be a meetconcave map such that for

every u v L with u v f u f v Let a be a joinirreducible of L If

P P

F a then there is a F a TL there is a decreasing matching TL

decreasing matching

P P

T L L F ha ai T L L F ha ai

f f

ai is joinirreducible in L Pro of Let L L L To check that ha

supp ose

ha ai hu v i hu v i

ai L Then u u a Using a f a f u u we obtain hu u

Since a v v

ai hu v i hu v i ha ai hu u

which yields u u a Since a is joinirreducibl e either u a or u a

Without loss of generality assume u a Then by denition of v a

ai as required so that hu v i ha

P P

Let TL F a TL F a b e a decreasing matching For

g TL F ha ai dene g by

g x h g x g x i

Since is decreasing g x g x Using h g x g xi L

and the assumption on f we get h g x g xi L Since g

and g are orderpreserving functions so is g Since g x a i

P P

x is in F it follows that maps TL F ha ai into TL F ha ai

Since is decreasing so is Using the assumption that is onetoone the

map g can b e recovered from g by

g h g g i

Hence is onetoone and it follows that is the required matching

Corollary Let f L Sub L be a meetconcave map such that for

every u v L with u v f u f v If L MPf P then L L

MPf P If L MPtP then L L MPtP

Example Let x b e a minimal member of Q and Q Q n fxg By

Q Q

Example L is a sub direct pro duct of L and L The asso ciated function

satises that u v if u v

Corollary Let Q be a poset If L MPf P then L MPf P If

L MPtP then L MPtP

This generalizes the part of Theorem which shows that every distributive

lattice ie every lattice of the form has the full matching prop erty

Neighborho o ds in Lattices

Though we now fo cus on lattices much of what will b e shown can b e

applied to semilattices by completion and use of Theorem

Let L b e a lattice By replacing L with the canonical unionclosed

F L of L if necessary Section we can assume that L is representation

a unionclosed family of sets such that L and L is generated by the join

irreducibles of L We then have and the domain of L is given by L

L L

Denition Let U L The lattice neighborhood in L of U denoted by

N U is the joinsubsemilattice of L generated by and the generators a

of L with a U

If U L u L and x u U then there is a generator a of L such

that x a u By the denition of lattice neighborho o d a N U This

yields the following observation

Observation If u L and U L then u U u U

N U

The map is the pro jection of L onto N U dened in Section

N U

A sequence of lattice neighborho o ds of U which includes increasingly

more distant generators of L is obtained by iterating N L

Denition Let N U N U For n dene

n n

U U N N N

L L

Intermediate lattice neighborho o ds are given by the ideals generated by

N U For n dene

n n n

U g U fV L j V N N U N

L L

Let N U fu L j u U g the ideal of L generated by U

Lattice neighborho o ds can b e used to characterize sub direct pro ducts

satisfying the conditions of Theorem

Theorem Let L and L be lattices The fol lowing are equivalent

i There exists a semilattice L and a meetconcave map f L

SubL such that L L L and for every u v L with u v

f u f v

ii There is a subset U of the domain of L such that L N U and

for every u v N U u U v U i u v

Pro of Assume i By Corollary we can assume that L is the internal

b e the pro jections and sub direct pro duct of L and L Let

L L

The map f is given by

f u f w j w L and w ug

Let U L n Supp ose that u v L and u U v U Let u

the assumption on L Then u u Since u u

L L L

f implies that there exists w L such that w u and w This

u hence u w u yields w w w u

L L

we obtain v u so that u v Note that v Similarly using u

L L

this implies that L L To show that L N U supp ose that a is

U L

a joinirreducible of L with a U Since a a a Since

a a a J L a a so that a L By arbitrariness of a

L N U Let N U L b e the map dened by u u U

L L U

Since u U u U for every u L it follows that y L is onto

N U

L This implies that is onto L so that N U L L

U U L U

The result follows

For the converse assume ii Let L b e the lubsubsemilattice of L

generated by the joinirreducibles of L disjoint from U Then L is the internal

sub direct pro duct of N U and L Let and We have

L L

N U

to show that if u u N U w L w u w v and u u then

there exists a w L such that w u and w v Such a w is given

by w w n U u Since w n U w n U w w v We have

w U w U u U so that by assumption on N U w u as

required

Observation Let U L The fol lowing are equivalent

i For every u v N U u U v U i u v

ii N U is a unionclosed representation of N U

L U L

iii There exists V U such that N V is an irredundant union

L V

closed representation of N V and N V N U

L L L

This implies that every sub direct pro duct of lattices L and L satisfying con

dition i of Theorem can b e obtained by the following construction Start

F L of L with domain with the canonical unionclosed representation F

U M L and an arbitrary unionclosed representation H of L on a dis

joint domain V Let G b e the joinirreducible generators of F Mo dify G by

adjoining to each nonempty member u of G a subset V u of V

G fu V u j u G g

Let F b e the unionclosed family of sets generated by G The unionclosed

family of sets F H is then a sub direct pro duct of the desired kind

The conditions of Observation are satised if for some element

x of the domain of L x is contained in exactly one generator a of L Then

the lattice neighborho o d of fxg is the twoelement chain f ag Theorems

and and the fact that MPf P imply that L MPf P for every p oset

P This can b e used to prove the following theorem

Theorem Let h be the height of L If h J L then L MPf P for

every poset P

Pro of Let u u u b e a saturated chain of L For each

i there is a nonempty generator a u such that a u Since h

i i i i

J L fa a g J L By construction there exists x a n a

h h i

Thus the only generator of L which contains x is a The pro of is completed by

the discussion preceding the statement of the theorem

Note that the pro of of Theorem shows that for every lattice L of

height h h J L

Lattices with Sp ecial Lattice Neighborho o ds

Let L b e a lattice and P a p oset

P P

Denition The decreasing matching TL F a TL G a is a

invertible i for every function f TL F a

f x a if x F

f x

f x otherwise

P P P

Observation If TL F a TL G a and TL G a

TL H a are ainvertible decreasing matchings then so is

Many of the commonly studied lattices have matchings which are ain

vertible see Theorems and b elow The existence of such matchings

is completely determined by a neighborho o d of a

P P

Theorem Let a J L and let TL F a TL G a be an

ainvertible decreasing matching Then the restriction of to T N a F a

is an ainvertible decreasing matching into T N a G a

Pro of Let I N a Then I is the ideal of L generated by the union of the

generators of L which intersect a As in Theorem the matching restricts

P P

to a decreasing matching TI F a TI G a If is ainvertible

then so is

The converse also holds

Theorem Let a J L and let L be a joinsubsemilattice of L such that

P P

L N a If TL F a TL G a is an ainvertible decreasing

matching then there is an ainvertible decreasing matching TL F a

TL G a

Pro of Let B fb J L j b a g Dene by

f x f x f x

for f TL F a and x P

f x a i Let f TL F a Since a N a L

L L

f TL F a so that f is welldened f x a This implies that

f whence f is order Since f is orderpreserving so are f and

B L

f x a which preserving Since f x a f x a i

f x we have f x implies that f TL G a Since

L L

f x f x for each x P so that is a decreasing map from TL F a

into TL G a It remains to show that is ainvertible Let x F Since

f x Therefore using the fact that L f x a is ainvertible

L L

is the internal sub direct pro duct of G B and L

f x f x f x a f x

L as required

Denition Let C b e a class of lattices and U L Then L is a

C neighborhood of U i L is a joinsubsemilattice of L such that L N u

and L C If C is the class of lower semimo dular geometric etc lattices a

C neighborho o d is called a lower semimo dular geometric etc neighborho o d

Denition The member u of L is lower semimodular i for every v w L

such that v covers u either v w covers u w or v w u w see Figure

d d d d

v w v w

d d d

u u

v w

d d

u w u w v w

Figure

Note that L is lower semimo dular i every u L is lower semimo dular

Theorem If L has a lower semimodular coatom c then for every a

J L with a c L a MPtP where the matchings are ainvertible

Observe that since J L there is at least one joinirreducible a L such

that a c

Pro of Let a L n a b e the map given by u u c for u a

Lemma The map is an orderpreserving decreasing matching and for

every u a u a u

Pro of The fact that is decreasing and orderpreserving follows by deni

tion The element covers c hence by lower semimo dularity of c and since

u c u u u covers u u c We have u u a u and since

u a u a u Therefore u a u This also shows that is

onetoone

Let F b e a lter of P Then induces a decreasing matching

P P

TL P a TL F a dened by

f x if x F

f x

f x otherwise

for f TL F a and x P The prop erties of imply that if f is order

preserving then so is f The identity u a u implies that is a

invertible

Corollary If L is a nontrivial lower semimodular lattice then L

MPtP

Pro of If c is a coatom of L then c is lower semimo dular and there exists a

prop er joinirreducible a c

Theorem If the dual of L is a geometric lattice and a J L then

L a MPf n where the matchings are ainvertible

Pro of The dual of L is geometric i L is coatomic and lower semimo dular

Every u L is the meet of the coatoms ab ove u This implies that for every

u a there exists a coatom u u with u a

To prove the theorem it suces to construct ainvertible decreasing

n n

matchings TL k n a TL k n a for k n The other

matchings are obtained by comp osition

Let f TL k n a Let f Dene f by

f k f k if i k

f i

f i otherwise

n n

for i n Then f TL k n a and f f in L By

k k

Lemma and since f k f k f can b e recovered from f

k k

f k f k if i k

k k

f i

f i otherwise

for i n Thus is as required

Theorems and yield

Corollary If a J L has a dual ly geometric neighborhood then L a

MPf n

Theorem If L is a geometric lattice and a is an atom of L then L a

MPf n where the matchings are ainvertible

Pro of As in the pro of of Theorem it suces to construct ainvertible

n n

decreasing matchings TL k n a TL k n a for k n

Let f TL k n a Let f Since f k a f k

f k Let A b e an indep endent set of atoms such that A f k see

Theorem Then A fag is indep endent Let B b e an indep endent set

of atoms such that B A fag and B f k Let B B n fag Then

W W W

B f k and B a Furthermore B a f k Dene f by

B if i k

f i

f i otherwise

n n

for i n Then f TL k n a and f f in L The map

k k

f can b e recovered from f by

f k a if i k

f i

f i otherwise

for i n Therefore is as required

Theorems and yield

Corollary If a J L has a geometric neighborhood then L a

MPf n

The Density Perspective

Let P and Q b e p osets

Denition The P size of Q is the cardinality of Q

The notion of P size is a generalization of cardinality The size of

Q is the cardinality of Q and the nsize of Q is the number of multichains of

length n of Q

of the If S is a subset of T then the density in T of S is the ratio

cardinalities of S and T More generally if we have a size measure w for the

members of a class of structures and S is a substructure of the structure T in

w S

the class then the w density in T of S is the ratio We formalize this for

w T

p osets and P sizes

Denition If R Q then the P density in Q of R is the ratio of the

P sizes of R and Q If x Q then the P density in Q of x is the P density

of the principal lter x generated by x If P is the oneelement p oset the

prex P is dropp ed

Let L b e a semilattice Let p Thus p is the number of lters

of P

If L has one of the downward matching prop erties for P then there

is a joinirreducibl e a L such that the P size of a is at most times the

P size of L In other words there is a joinirreducible a with P density in L at

most

then L a If the joinirreducible a of L has density in L at most

MPw The corresp onding statement for p osets other than may not hold

Denition The pair L a has the P density property i a is a joinirreducibl e

of L with P density in L at most The semilattice L has the P density property

i there is a joinirreducible a L such that L a has the P density prop erty

Whether all nontrivial semilattices have the P density prop erty is

unkown

The problem of nding b ounds for the minimum P density of the lters

generated by joinirreducibl es can also b e pursued from the following p ersp ective

Let C b e a class of semilattices Dene min and max

Denition For n dene

P P

h C P n min f L j L C and for every a J L fg a ng

Thus h C P n is the minimum P size of a semilattice in C with the prop erty

that the P size of each principal lter generated by a joinirreducible is at least

n For n dene

P P P

h C P n maxf L j L C and for every a J L fg L n a n g

hC P n is the maximum P size of a semilattice L in C with the prop erty Thus

that for every joinirreducible a L the number of orderpreserving maps f

P L of type other than P a is at most n

The functions h and h are related by

if h n n m then hm n m

hm n m then hn n m if

where the arguments C and P have b een omitted Since h and h are increasing

in the last argument this implies that

maxfhn j hn n mg hm m maxfn j hn n mg

hm n mg h n min fhm j hm n mg n min fm j

This do es not completely determine one function in terms of the other

Observation The fol lowing are equivalent

i Every nontrivial semilattice in C has the P density property

ii For al l n h C P n pn

hC P n n iii For al l n

To prove that ii implies i observe that if L is a nontrivial semilattice then

L p This implies that every nontrivial semilattice L with a joinirreducibl e

a ie a is maximal has the P density prop erty a such that

Let L b e the class of all semilattices Since every semilattice has at

least one element h L P hL P If L is a nontrivial semilattice

P P

and a J L then L n a p count the orderpreserving maps f

P f ag This implies that hL P n for n p

L then L has either at least two If L is a semilattice with

atoms or a nonatomic joinirreducible In either case there is a joinirreducibl e

P P

a with an element b L n a such that b The set L n a includes

p maps f P L onto f ag and p maps f P L into f bg Thus

P P

L n a p Using the twoelement lattice as an example we obtain

h L P n p for p n p

a then no maximal member If for every joinirreducible a L

of L is joinirreducible so that L contains a subset orderisomorphic to the

fourelement Bo olean lattice B This implies that L p Using B as an

L P n p for n p example we obtain h

In summary

Observation For n p

h L P hL P hL P n

For n p

L P n p h

and for p n p

hL P n p

The b est known general b ounds on h are given in Section

Before continuing the study of the P density prop erty and the functions

and h consider the function g dened as follows h

Denition For n dene

P P

g C P n min f L j L C and for every a J L TL a ng

C P n is the minimum P size of a semilattice L in C with the prop erty Thus g

that for every prop er joinirreducible a L the number of maps of type a

in L is at least n

The values of g for the class of semilattices are more easily obtained

than the values of h

Theorem Let n If P has a greatest element then

g L P n p

Pro of Let m d L P n m p consider e To show that g

the semilattices M of Example Since P has a greatest member the P size

of M is k p Let a b e an atom of M Then the complement of a

k m

is isomorphic to M Therefore the number of maps f M of type a

is mp n The P size of M is m p which is the

righthand side of the equality in the statement of the theorem Observe that if

T M m p n for every atom a of M a k m then

To show that g L P n m p supp ose that the semilattice

L has at least n maps f L of type a for every a J L Supp ose that

L has a nonmaximal atom a and let b a There are p orderpreserving

maps f P f bg such that b is in the range of f and p orderpreserving

maps f P f ag such that a is in the range of f Therefore the number of

maps f L not of type a is at least p This implies that unless L

is isomorphic to M for some k the P size of L is at least n p Since

n p m p the theorem follows

The [n]density Prop erty for Large n

Let L b e a nontrivial lattice

Theorem There exists m N such that for al l n m L has the n

density property

Pro of Let hP denote the height of the p oset P and let c P denote the

number of prop er chains of length i of P Let h hL

The pro of of the theorem requires the following result from the combi

natorics of p osets

Lemma Let P be a poset Then

P c P

The sum on the righthand side is known as the Zeta p olynomial of P see

Stanley

Pro of The chains of length i of P are the subsets of P isomorphic to i

Since the number of orderpreserving maps from n onto i is given by

and since the image of every orderpreserving map f n P is a chain

the result follows

By rewriting as P is a p olyno Lemma shows that

i i

mial in n of degree hP where the co ecient of n is given by

By Theorem we can assume that h J L otherwise L

MPf P for every p oset P and we are done

First supp ose that there is a joinirreducible a L such that ha

h Since the degree of a as a p olynomial in n is ha h while

the degree of L is h

lim n

Since the number of lters of n is n this implies that L a has the

n density prop erty for all suciently large n

Now supp ose that for every joinirreducible a L h a h

Then L is atomic so by assumption on L L has at least h atoms Every

chain C of length h of L consists of followed by a chain C C n fg of length

h of a for some atom a of L This shows that

c L c a

h h

aJ L

so that for some atom a L c a c L For this atom a is

h h

a p olynomial of degree h in n so that

h c a

lim n

c Lh h

Thus L a has the ndensity prop erty for all suciently large n

A Bound on the Minimum P density of Joinirreducibles

Denition A multisemilattice L consists of a semilattice L and a

multiplicity function L P For u L u is the multiplicity of u in

L For M L the cardinality of M is given by

u M

Note that every semilattice L can b e considered as a multisemilattice

L where u for u L

Let P b e a p oset and L a lattice Let p b e the number of lters of P

Let n If Theorem Let L be a multilattice with

P P P

n then M n where L n a L for every joinirreducible a L

M n is dened by

maxfk n p k p p

j k and p p ng if n p

M n

otherwise This bound is best possible

Asymptotically M n n log n Theorem

P P

Pro of Supp ose that for every joinirreducibl e a L n It L n a

M n L will b e shown that

L then for every prop er Consider rst the case n p If

P P P P

joinirreducibl e a L L n a L n a p count the number of

maps f P f ag with in the range of f Therefore given that n p

L L must b e the oneelement lattice and since M n

Now assume that n p Then M n n Hence we can assume

L We reduce the problem to the case where L is a Bo olean lattice that

Let M b e a minimal subset of J L with M Since L is non

trivial M Let B and k M Then B is a Bo olean lattice generated

by k atoms Let L B b e the restriction map dened by

u M u

Lemma The map is onto B

W W W W

Pro of Let N M We have N N and N N Consider

W W

N M n N

W W W W W

N M n N N M n N

W W W W

N M n N

W W

Minimality of M implies that N M n N M hence N N as

desired

P P

Since L B is orderpreserving induces a map L B

dened by

f f

P P

for f L The map is onto B To see this let B L b e the order

P P

preserving map dened by N N for N M If g B then g L

By the pro of of Lemma N N so that g g for every

g B

Let B b e the multilattice dened by

ff j f g g g

Since is onto g for every g B Since M we have u

L B

i u which gives Let Thus i f f

P P P P

L B L B

P P P

h B n fag then a M so that fag is an atom of B If h L and

P P P P

h L n a This implies that for every atom fag of B n B n fag

P P

B L Since the reduction is complete

It is now shown that we can assume every g B with g is a

P P

coatom of B The coatoms of B are given by the functions g where a is

the complement of an atom a B x is a minimal member of P and

a if y x

g y

otherwise

for y P

P P

Let C b e the set of coatoms of B Supp ose that g B n C and

g Let g b e a coatom with g g Let b e mo died to the

ax ax

multiplicity function dened by

if f g

f g if f g

f otherwise

P P P

for f B Then and since g g B B

P P

B B

P P P P

n B n b B n b for every atom b B

If we successively p erform this mo dication for every g B n C then

the resulting multiplicity function has the desired prop erty

Let m b e the number of minimal members of P Then C mk We

P k

B p Example Let r b e given by have

P P

r B B

k P

p B

P P

so that r is the total excess multiplicity of B Since every map f B with

f is a coatom

r C C

Let a b e an atom of B Since a is a Bo olean algebra generated by

P k P

a p If g C n a then g g for some minimal k atoms

x P This implies that the family fC n b j b is an atom of B g partitions C

into k sets of cardinality m Let

m sa C n a

so that sa is the total excess multiplicity of C n a

Let a b e an atom of B such that sa is maximal Then r k sa We

have

k k P P

p p sa n B n a

and

k k P

p r p k sa B

The rst inequality yields k sa k np p Using the second inequality

we obtain

k k P

p k n p p B

k n p k p p

Since k n p k p p M n this completes the pro of of the b ound

To see that the b ound is b est p ossible observe that if sa n

p p for every atom a B then equality holds in and for every

P P

n Thus M n for some k and B n a B atom a B

Let B B n fg where B is the Bo olean lattice generated by k

k k

P o k o P P

atoms Let B so that B p n B

k k

k k k

Theorem If P has a greatest member or k then p In

general p

P o P

Pro of Supp ose that P has a greatest member Then f B n B

k k

P o P P o

i f Thus B n B B where P P n f g Since the

P B P

k k k k

o k

number of lters of P is p p

Supp ose that P do es not have a greatest member Consider rst the

P o P

case k Since B the maps in B n B are determined by the

nonempty lters of P Thus p For k let denote the

k k

pro jection of B onto the ith comp onent of the representation B B

k k

such that for each i k there is Consider the family F of maps f B

P o P P o P k

an x P with f x Then F B n B Since F B n B

k k

F p

Theorem Let L be a multisemilattice where L is a semilattice

with no greatest member Let n If for every joinirreducible a L

o o P P P

n then M n where M n is dened by L n a L

o k

M n maxfk n p k p p k k

k k

j k and p p n g

k k

This bound is best possible

The requirement that k in the expression b eing maximized implies

that for n pp M n using the convention that max

This reects the fact that if L is a semilattice with no greatest member then L

has two incomparable joinirreducibles a and b so that

P P P P

L n a fa b g n a p p

Asymptotically M n M n n log n Theorem

Pro of Let L L f g b e the completion of L where is a new greatest

L L

element for L We have L We can p erform the reduction to the Bo olean

lattice B as in the pro of of Theorem using L instead of L We obtain the

o P o

multisemilattice B where B B n Let k b e the number of atoms

of B Since is not joinirreducible k

o P

Let C b e the set of maximal members of B Then C consists of the

maps g P B where a is the complement of an atom a B and g x a

a a

for every x P We can assume that if g then g C this follows as in

the pro of of Theorem using the maps g instead of the maps g

a ax

o o o P k

Since B B B p Thus the total excess multiplicity

r is given by

k o P o P o P

p B B B r

o P

Since every map f B with f is maximal

C C r

Let sa g Cho ose an atom a B such that sa is maximal Since

C n a fg g we have

o P P k k

B n a p p sa n

k k

C k we have r k sa which yields Since

o P k k

B p r p k sa

k k

The rst inequality gives k sa k n p p Using the

k k

second inequality we obtain

k k o P

p k n p p B

k k k

k n p k p p k k

k k

o P o

B Thus M n

If sa n p p for every atom a B then

k k

equality holds in This implies that the b ound M n is attained by B

for some k and

Theorem

M n n log n o

M n n log n o p

o o

Pro of Dene the functions f f c and c by

f k k n p k p p

o k

f k k n p k p p k k

k k

ck p p

o k

c k p p

k k

Then

M n maxff k j k and ck ng

o o

M n maxff k j k and c k n g

k o

Since p Theorem either one of ck n and c k n

implies that

k k

p p p p n

We show that implies k log n for n suciently large Supp ose that

k log n Then

k k k k

p p p p pp p p

log p

pp n n

k k

where we used the fact that p p is increasing in k Since p and

log p log p

p p

n on it follows that pp n n n for n suciently

large

Let k log n Then p k p p Thus we can

estimate f k by

f k n log n n n log n o

p p

For k f k n hence

M n n log n o

Using p we get

o log p

f k n log n n log n p n

p p

n log n o

which yields

M n n log n o

To show that the b ounds on M n and M n are asymptotically opti

mal let k log n log t where t is any sequence such that lim t

n n n n n

p p

and t on Then

k n n log n o

n p

and

n log n

k p o on log n

This implies

f k n log n on log n

p p

and

f k n log n on log n

p p

o k

on the constraints ck n and c k n are satised for Since p

n n

suciently large n The result follows

Let L b e the class of all lattices and S the class of all semilattices

Since L and S can b e considered as sub classes of the class of multilattices and

the class of multisemilattices resp ectively we have the following corollary of

Theorems and

Corollary

hL P n M n

hS P n maxM n M n

Neither b ound is exact For example supp ose that p n p

hS P n h L P n p Since By Observation

p pp p p pp

o k

it follows that M n If k and p p n then k so that

M n n p

Unionclosed Families of Sets Generated by Graphs

Denition The family of sets G is a graph i for every U G U The

members of G are referred to as edges The graph G is simple i for every U G

U For x G the degree in G of x is

G d x

fxg

If U is an edge of G the degree in G of U is

d U fV G n fU g j V U g

Denition Let F b e a unionclosed family of sets Let GF denote the family

of generators of F and dene J F GF n fg The closure in F of the set

X is given by

S S

F fV J F j V X g X

X F

This agrees with the notation for pro jections onto lubsubsemilattice s intro

duced in Section The set of isolated elements of X is

X X n X

F F

In terms of hypergraphs X is the set of isolated p oints of the hypergraph

F X

Observation Let F be a unionclosed family of sets The fol lowing are

equivalent

i U F

ii U U

U iii

In this section we will prove

Theorem If F is a unionclosed family of sets such that F and

J F is a nonempty graph then F has the density property

Let F b e a unionclosed family of sets such that F and let U F

Recall that N U the lattice neighborho o d in F of U is the unionclosed

family generated by the empty set and the members V of J F with V U

The third lattice neighborho o d is given by N U N N U

F F

To prove Theorem we develop a technique for estimating the

density of F in F This estimate dep ends only on the third lattice neighbor

U The estimate will b e used to show that if J F is a graph and ho o d N

U G fV J F j V g has minimal degree in G then F U has the

density prop erty Theorem

Denition Let H b e a family of sets The density in H of the set X is

Thus X is the density of H in H The recipro cal of X is denoted by

H X H

X Let U

H F

Denition The unionclosed family F is a conservative extension of F U

i there is a nonempty unionclosed family of sets H such that H U

and F F H

Asso ciativity of for families of sets yields

Observation The extension relation is transitive ie if F is an ex

tension of F U and F is an extension of F U then F is an extension of

F U

Since F F fg

Observation F is an extension of F U

Denition Let

U j F is an extension of F U g inf f

Observation

then F U has the density Observation If U J F and

property

The goal is to nd lower b ounds on in terms of the lo cal prop erties

of F at U To this end let F b e an arbitrary extension of F U and consider

Denition For X U let

X fY U j X Y F g T

We have

X F T

X F

and

F F

where equality holds if U F since F is closed under union with members of

F This gives

X T

X F

Note that if U F then this is an identity so that We will determine

X a lower b ound for T which is indep endent of F

Lemma Let X F If Y U satises

i X Y U Y

ii X Y X U n U

F F

then X Y F

Conditions i and ii are indep endent of F Observe that i is equiv

alent to

X Y U

and ii is equivalent to

X Y n U X U

F F

Lemma b elow gives conditions equivalent to i and ii in terms of the

family of generators of F

Pro of Since X F there exists Z U such that X Z F Since F is

an extension of F U there is a unionclosed nonempty family H such that

H U and F F H We have X Z A B for some A F and

B H We can assume that A X Z Let A X Y By i

F F

A U Y If we can show that X n A B then X Y A B F and

we are done

X Y n U By ii x X U n U Let x X n A Then x

F F

Since Z U X U X Z which yields x X Z n U Since

F F F

X Z n U X n A B we have x B as desired

Denition For every set X with X U let E X consist of the subsets

F U

Y of U satisfying the conditions of Lemma

E X fY U j X Y U Y and

F U F

X Y X U n U g

F F

Let E X E X

F U

Example Supp ose that F is the unionclosed family of sets generated

by the edges of the graph depicted in Figure and the empty set

Then

n o

J F fa bg fx ag fx ag fx ag fx bg fx bg fx x g

Let U fa bg We have

n o

E fx x g fbg fa bg

n o

E fx g fag fbg fa bg

n o

E fx x g fa bg

X for every X F Using inequal By Lemma E X T

ity we get

E X

X F

t t

S S

t t t S S

a b S

t t S

x x

Figure

Denition Let

E X

X F

F F

U Let

F F F

By inequality

U Observation

F F

We can dene E X in terms of the set of generators of F

Lemma The set Y is in E X i

i for every x Y there exists V J F with x V X Y

ii for every V J F if V n U X then for every x V n U there

exists V J F with x V X Y

Pro of In fact condition i of Lemma is equivalent to i and condi

tion ii of Lemma is equivalent to ii

If x W F then there is a generator V J F such that x V

W This implies that X Y U Y i i holds hence i i i

Supp ose that X and Y satisfy ii Let V J F and V n U X

Then V X U By ii X Y X U n U Hence if x V n U

F F F

then x X Y which implies that there is a generator V of F such that

x V X Y Thus ii holds

Conversely supp ose that X and Y satisfy ii We show that X

Y X U n U Let x X U n U Then there exists V J F such

F F

that x V X U We have V n U X so by ii there exists V J F

with x V X Y This implies that x X Y as desired F

Denition The neighborhood in F of a set X is given by

N X X fV J F j V X g

Let N N U and N N N U Observe that if U F then N

F F F

S S

N U and N N U

Lemma The family F is an extension of N U U

Pro of Let H b e the unionclosed family of sets generated by the empty set

and the generators V of F with V N U Then F N U H and

F F

H U This expresses F as the internal sub direct pro duct of N U

and H see Section

Theorem Let F be an extension of F U Then

U U

F F

N U F

Pro of By Lemma and Observation F is an extension of

N U is welldened By Lemma and by U U so that

N U F

denition of N U whether a given subset of U is in E X dep ends only on

the generators V J F with V N U Since J N U J F N U

F F F

Lemma implies that

E X E X

F U

N U

for every X disjoint from U The result now follows by denition of U

Denition Let D F An extension F of F U minimizes in D i

S S

F D and for every extension F of F U with F D

F F

Theorem Let D F There exists an extension F F H of

F U such that

D nU

i H is a lter of

ii F H

iii F minimizes in D

D nU

Pro of Note that i implies ii Let H b e a lter of Since F

F H H For the reverse inclusion let V F H Then there is a

W H with W V n U hence V n U H

Let F G b e an extension of F U such that G D n U and

F G minimizes in D Since F F G F G we have

nU nU nU

This implies that we can assume G F G

F G

F F G nU

D nU

Let H b e the lter of generated by G and let F F H We

which implies that F is as desired show that

F G F

For X G let

P X fY H j Y X g

Lemma Let X Y G with X Y Then P X P Y

Pro of Dene

Z Z n Y X

for Z P Y Since every Z P Y includes Y is a onetoone map To

show that maps P Y into P X let Z P Y Let X Z and

supp ose that X X Then X X By denition of Z X Y Let

Y Y X Then Y G and Z Y Y contradicting Z P Y

Lemma Let X G If Y P X then E Y E X

Pro of We show that for every Z U X Z Y Z Let Z U

F F

Let X X Z and Y Y Z The inclusion X Y implies

F F

X Y Since Y F and X G we have Y X F G so that Y X n U

Y n U X G Since Y P X and Y Y n U X it follows that

Y n U X Y X Using Y U Z we get Y X Z hence

Y X

The identities X Z Y Z and X U Y U

F F F F

imply that Z E X i Z E Y as required

For n let

G fX G j P X ng

By Lemma the G are lters of G Since F G G this implies

that if G then F G is an extension of F U such that F G

n n n

G Let N b e the maximum value of P X Then G G We use

n N

Lemmas and and the fact that the family fP X j X G g is a

partition of H to compute

E X

X F

E X

X H

P X E X

X G

P X

X G

P P

E X

X G nG

P P

X G nG

P P P

E X E X

X G X G

G G

Dene We have E X G and if G

F G F G n

X G

then E X G and This yields

F G n F G F G

X G

n n

G G

F G F G n

G G

F G F G n

G G

F G

G G

F G

which implies that F satises iii as required

Lemma Let X Y be sets disjoint from U If Y N X N then

E X E Y

Pro of By Lemma whether a given subset of U is in E Z dep ends

only on the generators of F included in N This implies that E Z dep ends

only on Z N

We strengthen Theorem

Theorem Let D F There is an extension F F H of F U

such that

D nU

i H is a lter of

ii F H

iii F minimizes in D

iv F fD n N g

Pro of By Theorem there is an extension F G of F which satises

i ii and iii Let H G fD n N g We show that F F H is as required

By construction F satises i ii and iv For X H let

P X fY G j Y N X N g

Then fP X j X Hg is a partition of G If Y P X then by Lemma

D nU

E Y E Y N E X N E X Since G is a lter of if X Y

then P X P Y Since P X fX N g for each X H this

nN nN N

implies that if X Y then P X P Y Let

H fX H j PX ng

Then H is a lter included in H for each n and H H Let N b e the

maximum value of P X By assumption on F G Supp ose

F H F G

that for some n Then

F H F G

E X

X G

F G

P X E X

X H

P X

X H

P P

E X

n X H

P P

n X H

n F H

n F G

F G

which is imp ossible Hence for each n F H also minimizes in D In

particular F minimizes in D as required

Corollary There is an extension F N U H of N U U such

F F

that

N nU

i H is a lter of

ii F H

U U iii if F is an extension of F U then

F F

N U F

Pro of Let D F and let F G b e an extension of F U satisfying the

conditions given in Theorem By Theorem F G is an extension

of N U U and

U U

F F G

U F G N F

By condition iv of Theorem F G is isomorphic to F G This

and Lemma imply that

U U

N U F G N U F G

F F

Since F is an extension of N U U F N U G for some nonempty

F F

unionclosed family G with G U The family H G G is a lter of

D nU

with H fD n N g Let H H Then

nN N

N U G G F G

N N

N U H

so that F N U H is the required extension

Corollary implies that to determine the minimum p ossible value

U F can b e replaced by N of U thus assume that F N U

F F

Henceforth we assume that U J F This implies that for every X

E X To show that F U has disjoint from U U E X so that

the density prop erty it suces to show that for every extension F of F U

Theorem The fol lowing are equivalent

i There exists an extension F of F U such that

N nU

ii There is a lter H of such that and for every min

F H

E X imal member X of H

Pro of Assertion ii is a sp ecial case of i Supp ose that i holds By

Corollary there is an extension F H of F U such that H is a lter of

N nU N nU

and Let H b e a minimal lter of such that

F H F H

Supp ose that there is a minimal member X of H such that E X Let

H H n fX g Note that the assumption on X and imply that

F H

N nU

H The family H is a lter of and

E Y

Y H

F H

E Y E X

Y H

E Y

Y H

F H

where we used the fact that if a b c and ac b then a bc

ac This contradicts the minimality assumption on H so that H is as desired

If there is no lter H satisfying the conditions in assertion ii of The

U for every extension F of F U so that F U orem then

has the density prop erty This yields

Observation If every extension F H of F U such that

N nU

i H is a lter of

E X ii for every minimal X H

satises then F U has the density property

F H

Let H b e an arbitrary lter satisfying i and ii of Observation

and let F F H We now assume that J F is a graph Thus for some

a b F U fa bg Assume that a b and let

N fx N n U j fx ag J F and fx bg J F g

Then N n U N N N see Figure

a b ab

pp p t t

b B

t t

p p

b p p B Q t t

N N

a b

p p

a b

Q t t

Figure

Lemma If X H then X N and X N

a b

Pro of Let X H Let X b e a minimal member of H such that X X

Supp ose X N Then either fbg E X if X N N or

a ab b

E X if X N N Since fa bg E X this contradicts

ab b

E X Thus X N By symmetry X N and we are done

a b

Corollary If X H and Y U then X Y U Y F

Pro of Let X H and Y U Supp ose that a Y By Lemma

there exists x N X so that fx ag F and fx ag X Y which implies

that a X Y Similarly if b Y then b X Y as required

F F

Denition Let Y U and x N n U Let E Y x consist of the subsets X

of N n U such that there is an edge fx y g J F with y X Y or y x

Dene

E Y E Y x

xN nU

N nU

Observation For Y U and x N n U E Y x is a lter of

Observation If X E Y then X and Y satisfy condition ii of

Lemma

Lemma If X E Y H then Y E X

Pro of Let X E Y H By Observation X and Y satisfy ii

of Lemma By Corollary X and Y satisfy i of Lemma

The result follows by the pro of of Lemma

Denition Let U b e a lter of Dene

Thus U is the density of U in

Theorem

X Y

U E Y x

F F

Y U

xN nU

Pro of By Kleitmans Lemma for every Y U

H E Y H E Y

Lemma and the fact that U E X for every X H yield

X X

E X fX H j Y E X g

Y U X H

fX H j U E X g fX H j Y E X g

Y U

H H E Y

Y U

Using H F we get

E X

X H

H H E Y

Y U

H E Y

Y U

E Y

Y U

Multiple applications of Kleitmans Lemma yield

E Y E Y x

xN nU

and the result follows

Dene

J F fV J F j V g

For x N n U dene

Ax ffx y g j y U and fx y g J F g

Ax Let nx

Lemma Let x N n U Then

i E x

ii If x N then E fag x and E fbg x

and E fbg x iii If x N then E fag x

iv If x N then E fag x E fbg x

Pro of Consider E x If fxg J F then E x If fxg

J F then

N nU N nU Ax

n E x

We have

N nU Ax N nU Ax

which implies that

E x

The remaining cases are proved similarly

Recall that d U is the degree in the graph G of U If U has minimal

degree in J F then F U has the density prop erty

Theorem If for every edge V J F with V U d V

J F

U d U then for every extension F of F U

F F

J F

Pro of We can assume that F F H where H is a lter satisfying i

and ii of Observation Let n d U n N n N and

a a b b

J F

N Then n

ab ab

n n n n

a b ab

Let x N We have d fa xg n and d fa xg n n nx

a a ab

J F J F

This gives nx n n n n n Similarly if x N then nx

a ab b ab b

n n Lemma and the assumptions on H imply that n and

a ab a

n By Theorem and Lemma

Y Y

E fag x E fbg x

xN nU xN nU

Y Y

nx nx

xN xN

n n

Let c It remains to show that and c

c c Without loss of generality assume that n n If n then

a b a

n n

b b

c c

m for m Using the fact that Supp ose that n Observe that

a ba for a and b Lemma we obtain

n n

b b

n c c

n n

b b

n n

as required

Lemma If a and b then a ba

Pro of The result is true for a Dierentiating b oth expressions relative

to a yields

d d

b b

a b a b ba

da da

The result follows

If J F then F is generated by oneelement sets so that F is a

Bo olean lattice Thus by Observation Theorem has b een proved In fact we have the following stronger result

Theorem Let F be a unionclosed family of sets and U J F Let

G fV J F j V U g If

i U

ii G is a graph

iii there is a simple graph G J F such that for every V G with

U V d V V G and d

G G

then F U has the density property

Pro of Supp ose that N is a unionclosed family such that F N N U

Then F is an extension of N U see the pro of of Lemma

Lemma shows that E X E X for every X with X U

F U N U

Therefore

U U

N F F F

for every extension F of F U Let N b e the unionclosed family generated

by G G fg Then F N N U Theorems and show

U for every extension F of F U It follows that F U has that

N F

the density prop erty

Notes

Section The theory of matchings is a well developed area of

combinatorics One of the earliest and b est known results is the theorem of

Konig and Hall It characterizes the families of sets F for which there

is an injective map f F F such that f U U for each U F Theorem

in Anderson

Decreasing and increasing matchings in lattices are discussed by Duf

fus and Kung Kung shows that if L is a mo dular lattice then

there is a decreasing matching M L J L thereby answering a question

p osed by Rival

The unionclosed sets conjecture Conjecture is stated in Duf

fus and app ears as an op en problem in Stanley Exercise pg

where it is called diab olical The problem is well known Winkler writes

that it is one of the most embarrassing gaps in combinatorial knowledge

Since the problem was p osed virtually no progress has b een made

toward its solution It is known to b e true for mo dular lattices R McKenzie

p ersonal communication and geometric lattices as mentioned in Duus

Neither of these results has b een published they follow from Corollary and

Corollary It is also known to b e true for small unionclosed families of

sets F F and and for some other sp ecial cases Sarvate and

Renaud and other unpublished work

Section Theorems and generalize some wellknown

elementary results related to the unionclosed sets conjecture A Ehrenfeucht

M Main p ersonal communications

Section The intersectionclosed representations of lattices ap

p ear implicitly in most textb o oks on lattice theory

Conjectures and are the usual way of formulating the union

closed sets conjecture

Section The sub direct pro duct is a useful generalization of

the direct pro duct Usually trivial sub direct pro ducts are excluded from the

denition The concept is discussed in Gratzer from a latticetheoretic

p ersective

Section Lo cality in lattices is usually dened in terms of inter

vals For example a lattice L is locally distributive i for every x y L such

that y is the join of the atoms of x the interval x y is a Bo olean lattice see

Greene and Kleitman

Section Most of the sp ecial classes of lattices for which the

unionclosed sets conjecture is known to b e true app ear in this section The

latticetheoretic pro of techniques used here are standard

Section There are many unsolved density related problems

Some examples follow

Problem Colb ourn and Rival see Sands Does there exist r

such that every nite distributive lattice L contains a joinirreducible a with

r r

Observe that Theorem shows that in a distributive lattice every minimal

and every maximal joinirreducible has joinirreducibl e has density at least

density at most

Problem Daykin and Frankl Is it true that for every convex sub

wC w

family C of

Note that is the density in P of a Sp erner antichain of P Daykin gives

a p owerful strengthening of Problem

Problem Daykin Which posets P have the property that for every

Q P the density of Q in P is less than or equal to the ratio of the number of

maximal chains of P intersecting Q to the number of maximal chains of P

Problem Knill Is it true that for every there exists w such

that if L is a lattice with wL w then there is a joinirreducible a L such

that Is this true for distributive lattices

is the widthdensity of a in L The ratio

The computations of Observation can b e continued for some other

small values of n For P JC Renaud has computed values up to n

unpublished

Section Bounds on the minimum value of m such that L has

the ndensity prop erty for all n m are implicit in the pro of of Theorem

but are quite large Lemma app ears in Stanley The Zeta p olynomial

nds many applications in the study of the combinatorics of p osets

Section Corollary and Theorem generalize the

known but unpublished b ound hL n C n log n

Section The class of lattices shown to have the density prop

erty in Theorem is a sub class of the class of distributive lattices The

distributive identity is given by

u u v u v u v

The distributive identity is given by

u u u v u u v u u v u u v

This is the dual of the nonequivalent identity given in Gratzer pg

Graphgenerated unionclosed families F with F satisfy the distributive

identity Pro of If a J F then a which implies that a u u u i

a u u or a u u or a u u Since u v fa J F j a u v g

the result follows

Huhn discusses ndistributive mo dular lattices

can b e generalized for P densities with P The estimates of

by using the generalization of Kleitmans Lemma to distributive lattices given

in Anderson pp However the resulting estimates are in general not

strong enough to prove the P density prop erty for graph generated unionclosed

families of sets

Computations similar to the ones in this section suggest that F U

has the density prop erty if for each V J F with V U d V is

J F

large enough compared to d U In some cases d V r log d U

J F J F J F

V j V J F g suces where r maxf

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