QUASIORDERS, PRINCIPAL TOPOLOGIES, AND PARTIALLY ORDERED PARTITIONS

VOL. 21 NO. 2 (1998) 221-234

QUASIORDERS, PRINCIPAL TOPOLOGIES, AND PARTIALLY ORDERED PARTITIONS

THOMAS A. RICHMOND

Department

ofMathematics

Western

KentuckyUniversity Bowling

Green,

Kentucky 42101

(Received August 26, 1996 and in revised form April 30, 1997)

ABSTRACT.The quasiorders on a setXareequivalent tothe topologies^on

X

which areclosed under arbitrary intersections. Weconsiderthe quaisorderson

X

tobe partialordersonthe blocks ofa partition of

X

and use thisapproach to survey somefundamental results onthelatticeof quasiordersonX.

KEY

WORDS

AND

PHRASES:Quasiorder,principaltopology, partiallyordered partition, specializationtopology,specialization order.

1991 AMS SUBJECTCLASSIFICATION CODES: 54A05, 54F05, 06F99.

1. INTRODUCTION

The study of topology on finite sets can be framed entirely in terms of equivalence relations, partitions,and partialorders.

A

naturalconstruction istopartiallyorderthe members

(blocks)

^of

apartitionofanarbitrary

(finite

or

infinite)

set X. Itiseasy to see thatsuchapartially ordered partition of

X

is equivalent to areflexive,transitive relation on

X,

that is, to a quasiorderon X.

In

1937, Alexandroff showed that quasiorders^on

X

^areequivalent to thetopologieson

X

in which arbitrary intersectionsof opensets areopen. Such topologiesarecalled principaltopologies.

For

Hausdorff topological spaces, theassociatedquasiorderrelationisequality. Theseconnections betweentopology anddiscrete mathematicshave not beenaswidely recognizedasthey deserveto be, largely duetothehistoricalconnectionsbetween topology andanalysis, withthe latter branch being primarily concernedwith Hausdorff topologicalspaces. Non-Hausdortt topological spaces arefindingnumerousapplications todayinareasof computerscience.

(See

^{thereferencesin}

^Kong

et al.

[14]

^andKopperman

[15].)

Considering themaspartially orderedpartitions, wewillstudy the principal topologies(quasiorders)^{on a}set

X,

withparticularattentiontothelattice structure ofthe collection of all principaltopologiesonX. Thiscollectionisthe collectionofall topologies on

X

ifXisfinite.

A

quasorder

(or preorder)

^{on a}set

X

is areflexivetransitiverelation onX.

A

partial order on

X

isan antisymmetricquasiorderonX.

A

totalorder on

X

isapartial order

_<

on

X

in which every pairof elements x, y E

X

arecomparable, that is,either x

<_

y ory

<_

x. The discrete order on a set

X

is

Ax {(x,x)

^{x E}

X}. (Thus,

the discreteorder relation on

X

is equality.)

A

symmetricquasiorderon

X

is anequivalence relatzonon X. If

(X, <)

^{is an} (partiallyor quasi-) ordered set, the decreasing hull ofasubset

B

C_

X

isdefinedtobe

dx(B) {x X

x

<

bfor

someb

B}.

^Wesay

B

isa decreasing set inX ifB

dx(B).

Increasing hulls and increasing sets aredefined dually.

A

lattzce is apartiallyordered set inwhich everypair ofelements has asupremumandaninfimum.

A

posetin whicheverynonemptysubset hasasupremumandan infimumis acompletelattice. Ifalatticehasaleastelement,^itwillbe called thezeroof the lattice, and denoted0. Similarly, if a latticehasagreatest element,itwillbe denoted1.

A

parttzon^ofa setXisanonemptycollection ofmutuallydisjoint, nonemptysubsetsofXwhoseunion isX. The members ofapartition

P

are called blocks ofP. Thereis anaturalcorrespondence betweenan equivalencerelation on

X

andapartitionofX: Theequivalenceclasses ofanequivalence relation

R

form the blocks ofapartition

P,

andapartition7⁹determines an equivalence relation

R

given byx

Ry

ifandonlyifx and yareinthesameblockof 79. The equivalence class ofa Xwillbe denoted by

[hi.

The complement ofaset

A

willbedenoted

A c.

If 79is apartitionofaset

X

and is apartialorder on79,wecall

(79, )

^apartzallyordered partition, orpopartztzonofX.

For

example,let

P

bethe partition

{[n,

ⁿ

+ 1)

^nisan

integer}

^of

therealline

R,

andorder 79 byagreeingthat

In,

ⁿ

+ 1) ___ ^[m,

^m

^{+ 1)}

^{ifand onlyif ngminthe}

usual orderon R. Then

(79, __)

isa partially orderedpartitionofR.

(In

fact, since is atotal orderon 9,

(79, __)

isatotally ordered partitwnof

R.)

Suppose

<_

is aquasiorderonX. There mayexist distinctelementsa,b Xwitha

<_

b and b

_<

a. Define a relation

R(_<)

on

X

by

aR(<_)b

ifandonlyifa

_<

band b

_<

a. Itiseasy tocheck that

R(_<)

^{is an} equivalencerelation on X. Let 79be the correspondingpartition. Forblocks

A,

B 79, put

A _ ^B

^{if andonly ifa}

^_<

^{b for some a}

^A

^{and some b B.} Equivalently, for blocks

[a], [b]

79, put

[a] - ^[b]

^{if andonlyifthereexist}

^{a’ [hi,}

^b’

^[b]

^with

^{a’ <}

^{b’. Now is}

apartial orderontheblocks of79.

In

thismanner,anyquasiorder

_<

on

X

generates apartially orderedpartition

(79, )

^ofX. Forexample, the quasiorder _E defined^on

R

byx_E yif andonly if

Ix ^{< [yl}

^where

Ix

^{denotes thegreatestinteger less thanor equal toxinthe} usual order

_<

on

R,

generates thetotally orderedpartition

(79, _)

described in thelastparagraph. On the otherhand, if

(7

9,

_)

^{is a}partiallyorderedpartitionof

X,

the relation

_<

definedon

X

bya

<

b ifand only if

[a] [b]

^{isaquasiorderonX. Thus, from}any partially ordered partition of

X,

we canobtain aquasiorderon

X,

andfromanyquasiorderonXwe canobtainapartially ordered partition onX. Performing these two tricks successively would carryusfromapopartition ofX to apopartition of

X,

^orfromaquasiorderon

X

to aquasiorderonX.

In

either case, the original structureisidenticaltothefinal one. Thus,our notionofpopartitionisidentical tothenotionof quasiorder,withonlyaslight disguise.

In 1937, Alexandroff

[2]

noted another occurenceof quasiordersinslightdisguise. Quasiorders on a finite set are identical to thetopologiesonthat set!

A

princzpaltopology ^on aset

X

is a topologyon

X

that isclosed underarbitrary intersectionsofopen sets. Observe that anyfinite topology, andin particular,anytopology^onafiniteset,^{is a}principal topology. Quasiorderson an arbitrary set areidentical to the principal topologiesonthat set! If

"

^{is atopology on}

^X,

the specialization order

<_

on

X

is defined by x

_<

y ifand only if x

cl({y}),

that is, if and only if every open neighborhood of y contains x. Itiseasytocheckthat the specialization order

_<

is aquasiorder. If

_<

isa quasiorderon

X,

theset of all decreasing subsets of

X

is a principal topologyon

X

called the speczahzatwn topologyassociated with

_<.

Sincecomplements of decreasingsetsin aquasiorderedset

(X, <_)

^areincreasing;sets, theclosureofasetBC_Xrelative tothe specialization topologyisthe smallest increasing subset of

X

thatcontains B. Thus, the increasing hull

ix(B)

^of

B

in

(X, _<)

istheclosure of

B

relative tothe specializationtopology.

A

goodaccountof principal topologies appearsinSteiner

[22].

Principal topologiesarecalled

Alexandroff

^{dzscrete topologies or} A-topologies in

Ern

and

Stege [9].

Principal topologies are precisely the topologiesin whicheverypointhasasmallest neighborhood. Karimpour

[13]

^defines

a complementary topologytobe atopologyinwhichthe complementofevery open setis open.

Example 5inSteenand Seebach

[21]

considers partitzontopologies,i.e.,topologieson aset

X

that haveabase consistingofa partitionofX. Complementarytopologies andpartitiontopologiesare identical,andareexamples of principal topologies.

A

topology^{Tona set}

X

isa

To

topologyif andonlyif for every pair of distinct pointsx,y E

X,

there isaneighborhood ofoneofthe points thatexcludesthe other point. Equivalently, v is a

To

^{topology on}

^X

^{if andonly iffor every pairof}distinct points x,y E

X,

one of the points is not in the closureof the other. If the topologyr isthe specialization topology associated with a quasiorder

_<,

the definitionbecomesris

To

^{if and}only ifforevery pairx, y

X,

either x i(y) ory

i(x),

i.e., either x yory x. Thus,wehave that the specialization topologyassociated withaquasiorder

_<

is

To

^{ifand onlyif}

^_<

^isantisymmetric, i.e., ifand onlyif

_<

isapartialorder.

An

excellent compilationof topologicalpropertiesand thecorrespondingpropertiesofthe asso- ciatedspecializationquasiorderisgivenin

Ern

andStege

[9].

Onecanfindthere, forexample, thataquasiorderon a finitesetis anequivalence relation if andonlyif the specialization orderis

2.

ELEMENTARY

LATTICE PROPERTIES

Givenaset

X,

therearenaturalpartialorderrelations ontheset

Q(X)

ofall quasiorders on

X,

onthe set

PoPar(X)

^ofallpopartitionsof

X,

andonthe set

PrTOP(X)

of all principaltopologies onX. Withtheirnatural order, eachsetisacomplete lattice, and theone-to-onecorrespondences betweenthesesetsdiscussedaboveare latticehomomorphisms. Belowwewill definethese natural orderrelations,andexaminethe lattice stucture primarilyby consideringthe lattice

PoPar(X).

Lattices ofprincipaltopologiesandquasiorderlatticeshave beenstudiedby

Ern [8],

^Steiner

[22],

and

Tma [27].

^{Manyof theworks onprincipal}topologies andquasiorders focusonapplications tofinite sets, ^{in which case} thelattice

PrTOP(X)

of all principal topologieson

X

issimply the lattice

Top(X)

of all topologies^onX.

Fortopologies’1,T2

Top(X),

^wesay₁

_<T r

^{ifand onlyif}’1 C_ T^assubsetsofthe power set ofX.

(We

^would say _{T1 is} coarser than

-.,

or ’2 is

finer

^than

’.)

Now

_<To

^{is apartial} orderon

Top(X),

and

(Top(X), <_T,)

^{is acompletelattice.}

(PrTop(X), <_T)

^{isalsoacomplete}

lattice, thoughgenerally the supremum andinfimumofaset in

PrTop(X)

may notagreewith the supremumand infimumofthe same settakenin

Top(X).

Forquasiorders

, Q(X),

^we

say

_<Q

^ifandonlyif

81

^::)

,

i.e.,if and only ifthe identitymap

idx (X, ) (X, )

isincreasing.

(We

use reverseinclusionhereasthe order on

Q(X)

^sothat

PrTop(X)

and

Q(X)

willbelatticeisomorphic, asopposedto latticeanti-isomorphic.)

Todescribe the partial order

_<

on

PoPar(X),

^we first consider the poset

Par(X)

^{of all}

partitionsofX. For

P, Par(X),

^wesay

P _<

ifand only if

[a]Q

^C_

[a]

for every a E

X,

in which case wesay is

finer

^{than 7}

p, _refines

7

p,

or

P

is coarser than

.

^{With this order,}

Par(X)

^{is a completelattice}

(Theorem

^{1.5 ofOre}

[18]).

Define a relation

_<

on

PoPar(X)

^by

(P, ,) _< (, Q)

^ifandonlyif

P <_

in

Par(X)

^andthequotient functionf,p

(Q, ) (P, Z,)

definedby

f([a])= [a],

^isincreasing.

PROPOSITION 2.1.

_<

is apartial orderon

PoPar(X).

PROOF. Antisymmetry: If

(P, ,) _< (, )

^and

(Q, Q) _< (P, ,),

then asmembersof

Par(X),

7^a

,

^andsincethe quotient functions

f,,

^and

f,,

^arebothincreasing, theorderon 7 mustagreewiththeorderon

,

^so

(:P, p) (Q, )

^asmembersof

PoPar(X).

Transztzvity:

If

(79, ,) < (, _--<)

and

(, ) < (7, r),

^then

P

in

Par(X),

and thecomposition of two increingnctions,

], f,m

^o

f,

^isincreing. Reflevity:

P P

in

Par(X)

^dthe

identitymapis increing.

PROPOSITION 2.2. If

((Pa, o))aeA

^{is a} collection of popartitions of

X,

then

VoeA(Pa, a)

ⁱⁿ

PoPar(X)

exists; itsunderlyingptitionis

P VaeA P

^(supremumin

Par(X))

anditsorder is

ven

^by

^{[a] [b]}

^{if donlyif}

^[a] [b]

^{forevery a A.}

PROOF. Itisey to

veri

^{that is a}partial order. With

P

described,

Pa P

d

f,

^isclelyincreingfor y

A,

^so

(P, )

isanupperbodof

((P, a))aeA-

^Suppose

(, )

^is

so

^anupper bound of

((Pa, ))aea.

^Then elements of

Par(X), P .

^Tosee

(P, )

^isthelet upper bod of

((Po, ))aeA,

^itonly

remns

toshow that

f,

^isincreing.

Suppose

[a] [b].

^Since

(, ) (P, a)

^for

eve

^a

A,

wehave

[a] a [b]

^forevery

e A,

andthus

[a] [b].

^U

A

posetin wch

eve

nonempty subset h asupremis calleda V-completesemilattce.

Any

v-complete semilattice

P

th alet element is acomplete lattice, for it is eily shown that for any S

P,

the supremumofthe set of lower bounds ofS is itselfa lower bound of Sand isthus the imum ofS. The dualresult for A-completesemilattices holds well. By

Prposition

2.2,

PoPar(X)

^{is a V-complete} semilattice. It h asmallest element, namely the indiscrete ptition

{X}. Ts

proves thefollowing results.

COROLLARY 2.3.

PoPar(X)

is acompletelattice.

A

descriptionoftheim ofaset

((P, a))aeA

ⁱⁿ

PoPar(X)

will beuseful. In

Par(X),

Aaea Pa

^canbe described

(see

p. 579 Ore

[18])

the ptition ofX whose blocks e the mimalchincoectedsubfiliesof

aeA a

^{where afaly is chain}connected ifforany

A,B e ,

^thereest

A, (i 1,...,n)

with

A A,

0,

A, A,+

0, d

A, B 0.

Viewing the partitions

(P)eA

^{eqvalencerelatio}

, _{AeA P}

can be described by the relation a b ifandonly if thereestsachainofequivMencesa

o

^{a, a,}

.

^a,+,

^a

^{b for}

some o,

A (i

0,1,...,

n).

Wewill saya,b

X

e loop connect in

OA Pa

^ifthereexist

a,

A

da,

X (i 0,1,...,n)

^tha0

a

^{a, and b a,forsome and}

[a,],., , [a,+],,,

^{for 0,1,...,n 1.}

^(2.1)

Thatis,a, b6

X

eloop connotedⁱⁿ

UA P

^{if thereis increingloop}

om [a],o

^to

[a],.

which psesthrough

Ibid,,

^forsomei.

^Note

^{thatifaischMnconnecttob,then thechinfrom}

ato bfollow by thechin

om

b toashowsthata and beloop connectedsimplyby ting

a,

ⁱⁿ

(2.1)

^tobe equMityatehlink. "Isloop cotedto" isan

uilence

^relation

P

onX.

Dee on

P

by

[a] [b]

^ifand

oy

ifthereestsachin

(2.1)

th

ao

^{ad b.}

Thisdefines a ptiM order on P. Since chinconnectedness implies loop coectedness,

P

is coserthan

Aaen ^P

^{where theiistenin}

^Par(X). om

theconructionof

(P, ), P

istheestpaitionbelow eh

P,

d

P

w given the nal ordernecessyto

me

it a lowerboundof

((P, a))eA.

Thus,in

gogar(X), (P, ) AA((P, )).

Note thatthe pitionderlying

AA((Pa, a)) (i

in

PoPar(X))

^{is not} generMly the se

AeA Pa (ium

ⁱⁿ

Par(X)).

Forexple, if

(Pm, ) ({A,A},A A )

^and

(P2, ) ({A, A}, ^A

2

A)

e poptitioof

X,

the im oftheirunderlyingptitions in

Par(X)

^is

{A,A},

^{whiletheinfimum in}

PoPar(X)

^is

({X}, A). However,

^fromthe description of supremaand

a

ⁱⁿ

PoPar(X),

^weseethat forscretely orderedpoptitions, theptitions underlying suprema andima in

PoPar(X) aee

^{withthesuprema dima}of the underlying partitionsin

Par(X).

^{Thuswehavethe}follongresult.

PROPOSITION2.4.

Par(X)

^{is asublatticeof}

PoPar(X).

It isshown in Pudlkand

Tma [19]

that "everyfinite lattice canbe embedded in afinite partition lattice". WithProposition 2.4,this impliesthat everyfinitelatticecanbeembeddedin

PoPar(X)

^{forsomefinitesetX.}

3. ATOMS AND COVERINGS IN

PoPar(X)

We

say b covers a in aposetXifa< ^b,anda

_<

c

<

b impliesc6

{a, b}.

^IfXisalattice with smallest element 0,anatomisanelementthat covers 0. If

X

hasalargest element1,acoatomisan element covered by1. The smallest and largest elements of

PoPar(X)

^aretheindiscrete partition

{X}

^andthediscretely ordereddiscrete partition

({ {z}

^"z6

X}, A),

respectively. The atomsof

PoPar(X)

arethetwoelementchains in

PoPar(X),

that is, thepartitionsof form

{A, A c}

^with

A -< A

or

A -<

A. Thecoatoms arethe discrete partitions withorders of the form

Axu{(Xo, Xl)}

where_x0and_xlaredistinctelements ofX.

Every

nonzeroelement of

PoPar(X)

^{isaboveanatom:}

For

(P,-<)

6

PoPar(X),

let

A

6

P

and consider

d,(A) {B

⁶7^9.

B

-<

A

in

(79, )}.

^Nowif i,(A) ^{isdefineddually,}

d,(A)

i,(A) ⁷⁹would imply

A

is maximum andminimum in

(79,-<),

so

(P,-<) ({A}, =)

^wouldbe the zero elementof

PoPar(X).

If

(79, _--<) #

0, assume,withoutloss of generality, that

d,(A) # P

^sothat

d,(A)

^and

\ d,(A)

^aredisjoint nonempty sets, the former decreasing andthelatterincreasing, that partition

"P,

and thereforepartitionX.

(79, _--<)

^isabove theatom

PA --= ^({d;(A),

⁷⁹

\ d,(A)}, d,(A)

< 7^=,

\ dp(A)).

Indeed,every nonzero popartitionis the supremum of theatomsbelowit: If

(79, Z)

6

PoPar(X)

^and

A

679isnot maximum,

PA

^gives

an atombelow

(:P,-<),

^andif

A

is not minimum,

({i;(A), P \ i;(A)}, i,(A) >

⁷⁹

\ i,(A))

^=_7 givesanatom below

(79, _).

Weclaim

V({79A A

6

T’, A

not maximumin

79}

^V

{

notminimumin

79}) (79, Z).

Theatomsinvolvedintheexpressionabove onlypartitionXinto partitionsoftheblocks of 79, sothe supremum ofthese atomswill giveapartition coarser than

"P. However,

sincepAV

7A (or

^{simply theonethat}exists, ifboth donot

exist)

gives a partition of

X

with

A

as ablock,we seethat the partitionunderlyingthe supremum of atoms given above issimply79. Suppose

[b],

^and

[c]

^areblocksin

P

with

[b] ^_< [c],.

^Each atom includedin the supremum abovepartitions 7 intoanincreasingset

I

andadecreasingset D. If

[b] I,

then

[c],

Iso that

[hi [c]

ⁱⁿany oftheatoms listed. Similarly,

[b] [c]

^if

[c],

^D.

In

the remaining case,

Ibis,

6

D, [c], I,

since

D < I

ⁱⁿeach atom,wehave

[b] ^_< [c]

in, eachatom. Conversely, if

Ibis, : ^[c],,

^{then for}

^{B [b]}

^wehave

^[b],

^B

^[cluB.

^{Thus,theorder on}the supremum of atoms givenabovedoes agreewiththe order

_

^on7

.

Beforeinvestigating coverings in

PoPar(X),

observe that

P

coversQin

Par(X)

^{ifand onlyif}

isobtainedfrom

P

by combining two blocks of 79.

For

coverings inthecollection

Po(X)

^{of all}

partial orderson

X

ordered byreverseinclusion

(i.e., (X, _<1) _< (X, _<2)

if andonlyiftheidentity function id"

(X, <_1) (X, <_2)

isincreasing),wehave the following result.

LEMMA

3.1. 8covers

inPo(X)

^ifandonly

if OU{(a,b)}

^forsome pair

(a,b) X2\8.

PROOF.Suppose 8covers

.

^{Then 8C}

.

^Let

^{(a, b) \}

^{8. Now}

a, b _<0

y}

isapartial order with8C C

.

Since 8 covers

,

^wehave

,

^{and thus,ifthere}

existsanotherpair

(c, d) \

^8,

(c, d) :/: (a, b),

then c

_<e

aand b

_<e

d, andatleastoneofthese inequalititesis strict. Now

’

⁸⁽³

^{{ (x,}

^y)"x

^<_

^c,d

^{_< y}}

^{is apartialorderwith8C}

’

^{C b,}

contrarytothefactthat8covers

.

^{Thus, if8covers}

^,

^then8C

^P

^and

^{[ \ 81}

^{1. Theconverse}

istrivial. ^[3

Sinceonly thecovering relations are depictedin the

Hasse

diagram fora poset,acharacter-

izationofthe covering relationsin

PoPar(X)

would be a useful tool. Although thefollowing result does not characterize the covering relations, ^itnarrowsdown the search forcoveringsand isstill veryhelpfulinunderstanding and describing thelattice

PoPar(X)

PrTop(X)

Q(X),

particularly whenXis asmall finiteset, aswewill seeinthenextsection.

PROPOSITION 3.2. If

(7 ,

-i<_,)covers

(Q, _--<)

in

PoPar(X),

thenoneofthefollowing conditionsholds:

1.

P

and

_--<e

^containsexactlyone moreorderedpairthan

_-<,.

2.

(,-<)

^isobtained by identifyingapairof blocks from

(P, _-<,),

one ofwhich covers the other in

(P, ,).

PROOF.

Suppose (P, _-<,)

covers

(,-<e)

ⁱⁿ

PoPar(X).

If

P ,

^then

^-<,

^covers

_--<e

in

Po(B)

^{whereB is theset of blocks in}

P ,

^{and the} lemma above implies that

__

^con-

thins exactly one moreordered pair than _{_-<9.} Now suppose

P # ,

^{that is,}

< :P.

^Sup- pose that

P

does not cover in

Par(X).

Then thereexist distinct blocks

[x],, [y],, [z],

in with

[x], -# [x], [y], # [y]e,

and

[z], # [z]e.

^{Without loss}of generality,

[x]e [y]e

and

[x], :, [y],

sothat

(n, ) {i,([x],),X \ i,([x],)}

^with

i,([x],)

^{as greatestelement isan}

atom below

(:P,-%)

ⁱⁿ

PoPar(X)

ⁱⁿwhich

[x]n # [y]n.

^{Ifthe blocks of}

T

do not split

[z],

i.e., if

[z]

C_

[z]n,

^{then in}

(S,s) sup{(,-<),(T,-<n)}

we have

[x]s # [y]s

even though

[x] [y],

and

[z]s [z] # [z],,

^sothat

(,, s)

^is strictly between

(:P, _--<,)

and

(,-<)

in

PoPar(X),

^contraryto

(P, ,)

covering

(, e).

^{If theblocks of}

n

do split

[z],

i.e., ifthere exist

z’,z" e [z]

^with

[x], -<, [z’],

^and

[x], :, [z’],,

^then

(T’, )= {i,([z’],),X \

with

i,([z’],)

^asgreatestelementisanatombelow

(T , -<,)

in

PoPar(X)

whichsplits

[z].

Since

[x],

-<,

[z’],

^and

[x], ;, [y],,

itfollows that

[x],, [y], e X \ i([z’],),

^{and thus}

[x], [y],.

Nowin

(S’,-’<’s) sup{(,),(7’,)}

^we have

Ix]s, [y]s,

even though

[x], :# [y],,

^and

[z’]s, # [z"]s,

^eventhough

[z’] [z"] [z].

Thus,

(S’, )’

^{is strictlybetween}

(P, ,)

and

(, e)

in

PoPar(X),

contrary to

(P,_-<,)

covering

(,_--<e).

Thus, if

(P, _--<,)

covers

(, )

in

PoPar(X)

^and

^P # ,

^{there can} not exist three distinct blocks

[x],, [y],, [z],

in

P

with

[x], -# [x], [y], -# [y],

^and

[z], -# [z].

Hencethepartition is obtained from

P

by iden- tifying twoblocks

A, B

of

P,

i.e.,

:P

covers in

Par(X).

Finally, weshow that

A

and

B

are related in

(7 , ,).

If they are not related, let

(7, _---<n)

^{be the}

popartition

with

T P

and with

8n ,

^t2

{{C, D}

^C

-, A,B

-Q,

D}.

^Now

(n, _-<n)

^is strictly between

(, _9)

and

(,___e)

in

PoPar(X),

again contradicting that

(P,-<,)

covers

(,-<).

Furthermore, since

(,-<e) ^_< (T’,-<,)

^{and is} obtained form

P

by identifyingthe two related blocks

A

and

B,

theseblocksmustformaconvexsetin

(P, ,),

i.e.,

A

covers

B

orconversely.

4.

FINITE

POPARTITION LATTICES

Withtheaidof Proposition 3.2, weareable to explicitly describe thelattices

PoPar(X)

^where

X

has2 or3 elements. Recall that when

X

isfinite, the lattice

Top(X)

of topologieson

X

^{is iso-}

morphic to

PoPar(X).

^{Wewilldenote ann-elementset}

{1,

2,...,

n}

byn.

Let Sn,k

represent the numberofpartitionsofnwithkblocks. Thesenumbers,theSterling numbersof the secondkind, arediscussed inStanley

[20].

^Let

Pk

denote the number of partial ordersonk. The numbers

P

^for

k 1,2,...,14 are given inErnand

Stege [9].

Itiseasy toseethat

IPoPar(n_)l ’=1Sn,Pk.

Thevaluesof

IPoPar(n)l

are givenfor smalln inTable4.1. The numbers

tell howmanytopologies thereare on ann-elementset. Formorevalues ofthesequencesgiven in

Table 4.1,seeErn4 and

Stege [9].

^{The number}

[Par(n)[

^{ofpartitionsofan}n-elementsetiscalled then^t’ Bellnumber, andis givenby

r"--1 ^[=o(-1)

^k

(k) ^(r k)"/r!. ^More

informationonBell numberscanbefoundinStanley

[20].

n

P, [Po()[ [Par()[ ITop(a)l [PoPar(a)l

1 1 1 1

2 3 2 4

3 19 5 29

4 219 15 355

5 4231 52 6952

6 130023 203 209527

7 6129859 877 9535241

8 431723379 4140 642779354

9 44511042511 21147 63260289423

1 3 13 75 541 4683 47293 545835 7087261 10 6611065248783 115975 8977053873043 102247563

Table4.1

In Top(2) PoPar(2),

the discretepartitionof _2with discreteorder isthe largest element.

Thepopartitions

({ (

¹

}, {

²

} }, {

¹

}

-<

{2})

^and

({ {

¹

}, {2} }, {

²

}

-<

{

¹

})

^aresimultaneouslyatoms and coatoms. Theindiscrete partition _2isthe least element. Thelattice

PoPar(2)

^{isshown in}

Figure4.2.

Top(2_) PoPar(2)

Figure4.2

The29elementsof

Top(3) PoPar(3)

are givennumericlabelsinTable4.3 below. Toavoid confusion withthese labels,^{we will}consider 3_ tobe theset

{a,

b,

c}.

The lattice

PoPar()

^isshown

in Figure4.4. Noticethat thisdiagramconsistsofatop element

(labeled 1),

^{abottomelement}

(labeled 29),

^threestacked crowns,

({2,

3,...,

13}, {8,

9,...,

19},

^and

{14,...,

19, 23,...,

28})

^and

three other elements

(20,21,

^and

22). A

nice three-dimensional model ofthis lattice diagram can be constructed by drawing the crowns in different colors on aclear cylinder and carefully plaing the vertices20,21 and 22 insidethecylinder, avoiding unnecessaryintersectionsoflines

(or

^points).

Thecoveringsin

Top(4) PoPar(4)

^arealsoeasily obtained withtheaidof Proposition 3.2.

However,

since

PoPar(4)

^{has 355}elements,itsHassediagram becomes unwieldly. Theset

Po(4)

ofallpartial orderson4 elementsappearsattheupperend of

PoPar(4_)

^asthepopartitionsof 4_

whose underlyingpaitions are discrete. The

Hasse

diagram for

Po(4_)

has219verticesand 588 edges. We list 16 isomorphic classes of

Po(4)

ⁱⁿ Table 4.5, and indicate the covering relations

amongtheseclassesinFigure4.6. Theweights onthe edges of the graphin Figure4.6represent thenumber of elements from thelarger isomorphism class thatcovereach element from the smaller isomorphism class. Forexample, the edge from

III

toVIhasweight 3, indicatingthateach partial order on4of type VI is coveredby 3partial orders oftype III. These weights, together with the numberofpartial ordersofeach type

(from

Table

4.5)

^can be used to verifythatthe

Hasse

diagram for

Po(4)

^{has 588edges.}

1. Discrete partition with discreteorder

{a} {b} {c}

c

2.-7.

{x}

^| willbe denoted

a

{X} {y}

{

^xy

8.-13.

{z}

^or

^{{x} {y},}

^denoted z

ac a

10. ab

or z

11. b

12. bc

a 13. c

14.

X}

^X

14.-19. Chains

{y}

willbedenoted y

a b b ^c

15. 16. 17. 18. 19.

Two-bock

partitions * 20.-22.

withdiscreteorder

{x} {y, z}

20.

{a} {b,c}

^21.

{c} {a,b}

^22.

{b} {a,c}

23.

23.-28. Two-blockpartitions

{x} {x,y}

withtotalorder,

I ^{y,z}

^r

I ^{Z}

{}

{b,c}

=4.

{ } {b}

.{,}

^26.

^{b,c} {a}

^27.

{c}

{,b}

^28.

a,c}

{b}

29. Indiscrete partition

{a,b,c}

*

1

Elements of

Top(3) PoPar(3)

Table 4.3

Top(3) PoPar(3)

Figure 4.4

Isomorphism

Type

III

IV

VI VII VIII

X XI XII

Diagram

Number

ofp.o.s ofthistype

12 24 12 24

24 48

12 24

24

Typesof Nonisomorphic PartialOrderson4 Table4.5

CoveringRelationsin

Po(4)

Figure4.6

5. TOTALLY ORDERED PARTITION LATTICES

Theset

ToPar(X)

of totally orderedpartitionsof

X

isnotalattice if

IX ^>

^{1, forif}

IXI ^>

^1,the

totallyordered partitions

({A, AC}, A -<

^A

c)

emd

({A, AC},A -< A)

havenosupremum. Arbitrary infimaofmembers of

ToPar(X)

^existin

ToPar(X)

and agreewiththe

infi,

^{ma in}

PoPar(X).

^To

seethis, itsuffices toshow that if

(:Pa,-,) ToPar(X)

^forc

A,

then

(7 , _-_-<) =- ^{AaA(:Ps, ___a)}

istotally ordered. Given

[a],, [b], P, [a]

^and

[b]

mustbe related in

(Vso, "<so)

^{for each}a0

A

since

(7so,---<so)

^{is totallyordered.} Thus,thereis achainoflength 1as in

(2.1)

between

[a]

^and

[b],

^so

[a]

^and

[b]

^{arerelated in}

(7 v, ).

Thus,

ToPar(X)

^{is a}A-completesublatticeof

PoPar(X).

ThesetMof totallyordered discrete partitionsof

X

occupiesacentral position in

PoPar(X).

The elements of M are the minimal elements of

Po(X) {(P,,) PoPar(X)

^{7) is the}

discrete partition of

X},

^andtheyarethe maximal elementsin

ToPar(X).

Indeed,in

PoPar(X), d(M)

N

(M) M, d(M) ToPar(X)

and

i(M) . ^Po(X).

^{This last factsays that forevery}

partialorder

_<

^on

X,

there is a total order _<t on

X

such that id

(X, <_) (X, <_t)

is increasing; that is, every partialorder on

X

canbe extendedto atotal order

(see,

e.g., Davey and Priestley

[6],

p.26). We^alsonotethat

d(M)

N

i(M) M

isthe definitionof

M

being order convex. Equivalently,

M

isorderconvexif x

_<

y

_<

zand x,z

M

implies y M.

IfX n,it iseasy toseethattherearen!elementsof

M,

corresponding to the permutations ofn.

In PoPar(3)

^{with thenotationof Table 4.3}and Figure 4.4, theset

M

of totally ordered discrete partitionsisthe set

{14,

15,...,

19}, Po(3) . i(M) {1,

2,...,

19}, ToPar(3) d(M)

{14,...,

19,23,...,

29},

^and

i(M)

^t3

d(M) {1,

^2,...,

29} \ {20,

21,

22}.

^{Thelatticediagramfor}

the latter setcanbeobtainedfrom Figure4.4byomittingthe dashedlinesand thethreeassociated vertices.

Foranytotally orderedpartition

(’P, _--<,)

6

ToPar(X),

^{the set}

d(T’) {(Q, _-<)

6

PoPar(X) (Q, _--<) _< (P, p)}

^isaA-completesublatticeof

PoPar(X)

^withgreatest element

(7 , -<9),

and isthereforea completelattice. Themembers of

d(7)

^areprecisely the convex partitions of the totally ordered set

(:P,-<,).

For example, if

(P, _--<,)

is a totally ordered partition of

X

order isomorphic to n with the natural order, then

d(P)

corresponds to the convex partitions of__n.

There are2^"-1 of these

(there

are twice as many convex partitionsof k as ofk- 1--you can add a newblock

{k},

^orinclude k in

[k- 1]).

^Thus,beloweach of then! maximal elementsin

ToPar(n__)

are2^"-1 members of

ToPar(n_).

Of course, onemember of

ToPar(n_)

may be below severalmaximalelements,so wemayconclude

IToPar(n__)l <_ n!(2n-1).

Recalling that

S,,k

gives the numberof partitionsofn with kblocks and that therearek! total ordersonk, itiseasyto seethat

IToPar(n)l .’=l(S,,k)(k!).

The values of

IToPar(n)l

for small values ofnare given inTable4.1.

We may also generate thenumbers

IToPar(n)l

recursively.

In

an arbitrary totally ordered partitionof n, suppose the top block contains k elements ofn. There are

()

^{ways to choose}

these k elements. Theremaining n-kelementscan beformedintototally orderedpartitionsin

IToPar(n k)l

ways. Letting k range from1 ton,^wehave

IToPar(n_)l

Herewetake

IToPar(O)l

^{tobe 1.}

A

chain topology^on

X

is atopology^on

X

whose open sets aretotally ordered by inclusion.

ChaintopologiesareconsideredinDoyle

[7]

^andStephen

[23].

The proposition below may be used to translate the recursiveformula for

IToPar(n)l

^{in the preceding} paragraphintotherecursive formulagivenbyStephen

[23,

^Theorem

2]

^{forcountingthenumberof}chain topologies onn.

PROPOSITION 5.1.

A

popartition

(:P,-<)

of

X

is atotally ordered partition if and only if the specialization topology

-,

is achaintopologyonX.

PROOF. Forx 6

X,

let

N(x) CI{N

6 T, x 6

N}.

^Theresult follows easilyfrom the equivalence ofthesethree statements:

(1) N(x)

^C_

N(y), (2)

^y6

cl{x}, (3)

^y

^<_

^{z. [3}

Doyle

[7]

^{definesatower}space on n points to be anyspace homeomorphicto n withthechain topology

{@,

^1,2, 3,...,

_n_n}.

^Moregenerally,we willsayachaintopology,ron

X

is atower topology on

X

if forevery

U

6

"

^{thereexistsV 6}

-

^andx6

^{X \}

^VwithVt3

^{{x} U.}

Doyle showsthat afinitetopologicalspace

X

is aT0-spaceifand onlyifthereis acontinuousone-to-onefunction from

X

toatower space. Wewillshowthat thisisequivalent to the statement thataquasiorder onafiniteset

X

is apartial orderifand onlyifitcanbe extended toatotal orderon X. First, wewillgive awell known fundamental resultonfunctions.

PROPOSITION 5.2. If

(X, _<x)

^and

(Y, _<y)

arequasiordered sets with associated specialization topologiesTXandTr,respectively, then

f ^{(X, _<x)} (Y, _<r)

isincreasing ifand only if

f" ^{(X, TX) (Y, rr)}

is continuous.

PROOF. Suppose ]"

(X, _<x) (Y, _<r)

^is increasing. IfU is openin

(Y, rr),

then U

isdecreasingin

(Y, _<r),

andit followsthat

f-(U)

^isdecreasingin

(X, _<x),

i.e.,

f-I(U)

^is

rx-

open. Conversely, suppose

f ^(X, rx) (Y, zr)

^iscontinuous. Then, fromthe definitionof the topologiesinvolved, theinverseimageofany<_y-decreasing setis<_x-decreasing. Supposea

<_x

b yet

f(a) :r ^f(b).

^Then

^{f(a) dr(f(b)).}

^Butsince

f-*(dr(f(b)))

^{is a} _<x-decreasing set that containsb,wehavea 6

f-*(dr(f(b))),

contrary to

f(a)

^6_

dr(f(b)).

Recalling that the specialization order for a topology is a partial order ifand only ifthe topologyis

To,

we nowobserve that the followingstatements are equivalent:

(a)

^aquasiorderon nisapartialorderifand onlyifitcanbeextended toatotalorderonn.

(b)

aquasiorder

_<

onn is apartialorderifand onlyifthereisatotalorder _<tonnsuchthat id

(n__, <_) (n, <t)

isincreasing.

(c) A

topologyronn__is

To

^{ifandonlyifthereis atotalorder}

<t

^{on__n suchthat id}

(n_n_, "r) (_n, r_<,)

is continuous.

Noting that thespecializationtopology_< ^fromatotal orderon afinite set givesatowerspace, and that the identityfunction istriviallyone-to-one,we seethat the following statementisimplied bythoseabove.

(d) A

topology on_n is

To

^{ifand onlyifthereis aone-to-onecontinuousfunction}from__nonto atowerspace.

Since the function in

(d)

^is a permutation on __nandsinceany two tower spaces on __narehome-

onorphic,

we may assume, without loss of generality, that the function in

(d)

^is the identity.

Furthermore, since any towerspace topologyon__n isthe specializationtopology for sometotal orderon_n_n,statement

(d)

implies

(c).

^Thus,allofthe statements

(a)-(d)

areequivalent.

The lattice

PoPar(X)

^isisomorphic tothelatticePrTop(X)ofprincipaltopologiesonX. By Proposition5.1,

ToPar(X)

^{is isomorphicto}the principalchaintopologiesonX. Fortheset M of totally ordereddiscrete partitions of

X,

wehave

M i(M)

^fq

d(M) . Po(X)

^fq

ToPar(X).

Recalling thatthe lattice

Q(X)

of quasiorders on

X

^is isomorphicto

PrTop(X) .. PoPar(X)

and that the partial orders on

X

correspond to

To

^principal topologies, we see that M

Po(X)fToPar(X) {To

^{principaltopologieson}

X}fq{principal

chaintopologieson

X} {tower

topologies on

X}.

Thus, the set

T

^oftower topologies on

X

occupies a central position in

PrTop(X),

with

i(T)

isomorphic to the V-completesemilattice of

To

principal topologies on

X,

and

d(T)

^{isomorphic to}the A-complete semilattice of principal chain topologies on X. Again recall thatifXisfinite,alltopologiesonXareprincipaland wecouldomittheword "principal"

in thediscussionabove.

6. OTHER LATTICE PROPERTIES

In

this section, we will assumesome familiaritywith some standardlattice theory concepts as giveninBirkhoff

[4].

Backgroundmaterial oncontinuouslattices canbe foundinGierz et al.

[11].

PoPar(X)

^iscomplemented. Somecomplements of

(7

9,

)

E

PoPar(X)

are justthe discretely orderedcomplements of 79in

Par(X).

^{Thus, for}

(79, _)

^a__

PoPar(X),

ifwelet

A

beacomplete setofequivalence class representativesfrom 79,then thediscretely orderedpartition

{A}

^tA

{ {x}

x X

\ A}

^isacomplement of79. If

IX[ >_

3,thenthereexistpopartitions

(79, )

that are neither the topnor bottom elementof

PoPar(X)

^{andhave oneblock}with morethanoneelement ofX.

Forsuchapopartition79,the completesetof equivalence classrepresentativesfor79isnot unique, that is, 79is notuniquely complementedin

Par(X).

Itfollows thatif

IX[ ^_>

3, then theelementsof

PoPar(X)

^arenotuniquely complemented. Sincecomplementsare uniqueindistributivelattices, weseethat

Par(X)

and

PoPar(X)

^arenot distributive if

IX[ ^>_

3. While any interval in

Par(X)

iscomplemented

(Ore [18, p.596]),

thisresult doesnot holdin

PoPar(X),

for, e.g., in

PoPar(3)

(see

Figure

4.4),

27hasnocomplementin theinterval

[29, 13].

In

acompletelattice

L,

wesayx is way belowy, denotedx

<<

^{y iffor} any directed subset

D

C

L,

y

_<

sup

D

implies thereexistsd 6

D

with x

_<

d.

A

completelattice

L

is a continuous laiceif for everyx6

L,

x

sup{y L"

y

<< x}. In

afinite lattice x

<<

^yifandonlyif x

_<

y

(See

Remark 1.5, p.41inGierz et al.

[11]),

andthusevery finite latticeisacontinuouslattice. In particular,if

X

isfinite,

PoPar(X)

is a continuous lattice.

IfX isinfinite,then

PoPar(X)

^isnotacontinuous lattice. Toseethis, wewillshow that if

X

isinfinite, then

P <<

ⁱⁿ

PoPar(X)

^ifand only if9is the bottom element of

PoPar(X),

i.e., theindiscrete

(one-block)

partitionofX.

(For

simplicity, wewill supress the orders onall popartitions in this

paragraph.) Suppose X

isinfinite. EmbedacopyNofthenaturalnumbersin X.

For

eachz

X

andeachn

N,

define

Tz(n)

(respectively,

TZ(n))

^tobe the discrete partition of

X

withthe order

{z} ^<_ {m}

(respectively,

{z} ^>_ {m})

^{for all}m NC_

X

with m

>_

n. Define

D {T(n)

ⁿ 6

N}

^and

^D {T)(n)

^{n 6}

N}. Dz

^and

^D

^aredirected sets in

PoPar(X)

with

9(n) ^_< 9z(m)

^ifand only if

9Z(n) ^_< TZ(m)

^{if and}only ifn

_<

m. Thus,

D

and

D

^are

chainsin

PoPar(X)

^{isomorphictoN.}Nowforany z6

X,

sup D, sup

D

isthe top elementin

PoPar(X),

namely,thediscrete partitionof

X

withthe discreteorder. Thus, Q

_<

sup

Dz

^sup

D

forany Q6

PoPar(X).

For9

<<

^{Qtohold,wemusthave 9}

^_< T)z(nz)

and9

_< :D(rn)

for somemz,nz 6 Nandfor everyz6

X.

Forz 6

X,

define

T

{j 6N"j

>_

rn,j

>_ n}.

^Now

9

<<

Q implies 9 isa popartition with

[t,], <_ [z], _< [t]

for every z6

X

and every

t

6

Tz.

Thus, in9,everypoint zof

X

isidentified with a tail

Tz

^{of thesequence N.Sinceallsuchtails}

overlap,

P

isthe indiscrete partition ofX. Itfollowsthat

PoPar(X)

isnot continuous, sinceany nondiscrete popartition isnotthe supremum ofpopartitionswaybelowit.

Forany naturalnumbern,

Par(n)

^{is a}semimodular lattice. SincetheJordan-Dedekind chain condition

(all

maximal chainsbetweentwo given pointshave thesamelength)holdsinanysemi- modular poset offinitelength,

Par(n)

^isgraded byitsheight function

h(7)

^maximumlength ofachainfrom theindiscrete partition to9.

(See

^pp. 5,15-16,40of Birkhoff

[4]).

^Wewillshow

thatforn

>

2,

PoPar(n__)

^doesnotsatisfythe Jordan-Dedekind chaincondition, and thereforeis neither semimodular norgraded, forn

>

2,consider popartitionsPl,9,

93

^{allwiththediscrete}

partitionof n, andwith 1

<

^{2 in}91, 1

<

2,1

<

^{3 in}9, and 1 < ² < ³ in 93, with no other order exceptx

_<

xfor allx 6 n.

Let 94

and

9s

^bepopartitions whose underlyingpartitionis

{ {

^1,

2} }

^t3

{ {x}"

x 3,4,...,

n},

with

94

carryingthediscreteorder,and with

{

^1,

2}

<

{3}

ⁱⁿ

9.

Now 91,7, 93,

7:’

^and91,94,

9

^aremaximal chainsfrom

91

^{to withdifferentlengths.}

ACKNOWLEDGMENTS.The authorwouldlike toextendhisthanksto theGeneral Algebra workinggroupatTechnischeHochschuleDarmstadtfor their kind hospitality during the prepara- tionofthispaper, and to Prof. MarcelErn4at UniversitiitHannover.

References

[1] AIGNER, M.,

Beitrge zumVerband der Partitionen, Habilitationsschrift im Fachbereich Mathematik, Eberhard-Karls-Universitit, Tiibingen, Germany, 1971.

[2] ALEXANDROFF, P.,

DiskreteRiiume,Mat. Sb.

(N.S.)

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