€ro du™ing xew fije™tions from yld

@to —pp e—r in edv—n™es in w—them—ti™sA

epril PPD IWWR

h—vid peldm—n

niversity of xew r—mpshire

t—mes €ropp

w—ss—™husetts snstitute of „e™hnology

efƒ„‚eg„

por —ny purely ™om˜in—tori—l ™onstru™tion th—t pro du™es new nite sets from

given onesD — ˜ije™tion ˜ etween two given sets n—tur—lly determines — ˜ije™tion

˜ etween the ™orresp onding ™onstru™ted setsF ‡e investig—te the p ossi˜ility

of going in reverseX dening — ˜ije™tion ˜ etween the origin—l o˜ je™ts from

— given ˜ije™tion ˜ etween the ™onstru™ted o˜ je™tsD in ee™t ’™—n™eling4 the

™onstru™tionF ‡e present — pre™ise formul—tion of this question —nd then give

— ™on™rete ™riterion for determining whether su™h ™—n™ell—tion is p ossi˜leF sf

the ™riterion is s—tisedD then — ™—n™ell—tion pro ™edure existsD —nd indeedD for

m—ny of the ™onstru™tions th—t we h—ve studied — p olynomi—lEtime ™—n™ell—E

tion pro ™edure h—s ˜ een foundY on the other h—ndD when the ™riterion is not

s—tisedD no pro ™edureD however ™ompli™—tedD ™—n p erform the ™—n™ell—tionF

„he ™ele˜r—ted involution prin™iple of q—rsi— —nd wilne ts into our ru˜ri™

on™e it is re™ognized —s — thinly disguised form of ™—n™eling disjoint union

with — xed setD itself — well{known te™hnique th—t ts into our theoreti™—l

fr—meworkF ‡e show th—t the ™onstru™tion th—t forms the g—rtesi—n pro du™t

of — given set with some otherD xed set ™—nnot in gener—l ˜ e ™—n™eledD ˜ut

th—t it ™—n ˜ e ™—n™eled in the ™—se where the xed set ™—rries — distinguished

elementF ‡e —lso show th—t the ™onstru™tion th—t t—kes the mth g—rtesi—n

p ower of — given set ™—n ˜ e ™—n™eledD ˜ut th—t the ™onstru™tion th—t forms

the p ower set of — given set ™—nnotF

uey wordsX fije™tionD ˜ije™tive pro ofD qEsetD g—rtesi—n p owerD p ower set

IF sntro du™tion

emong the v—rious me—ns of est—˜lishing ™om˜in—tori—l identitiesD m—ny ™omE

˜in—tori—lists esp e™i—lly prize ’˜ije™tive pro ofs4X dire™t demonstr—tions ˜y

me—ns of m—nifest ˜ije™tionsF „he present st—tus of the s™ien™e of ˜ije™tive

pro of m—y ˜ e likened to th—t of rulerE—ndE™omp—ss ™onstru™tions prior to

q—uss —nd ‡—ntzelF iven with m—ny ingenious ex—mplesD ’˜ije™tive pro of 4

rem—ins —n inform—l notionF „he l—™k of suit—˜le form—liz—tion prevents exE

plor—tion of the limit—tions of ’˜ije™tive pro ofD4 —nd so no results we know of

m—y ˜ e interpreted to s—yD in —ny senseD th—t there exist —n—lyti™—lly veri—˜le

™om˜in—tori—l identities —dmitting no ˜ije™tive pro ofF sndeed the situ—tion

m—y well ˜ e sensitive to how ’˜ije™tive pro of 4 is m—de pre™iseD mu™h —s —nE

gles m—y ˜ e trise™ted with m—rked str—ightedgesD ˜ut not with unm—rked

str—ightedgesF

sn this p—p erD we show th—t just —s —lge˜r—i™ metho ds @—s em˜o died in q—lois

theoryA ™—n ˜ e ˜rought to ˜ e—r on questions of ™onstru™ti˜ility of geometriE

™—l o˜ je™ts ˜y use of geometri™—l to olsD so to o ™—n metho ds from —lge˜r— ˜ e

—pplied to questions of ™onstru™ti˜ility of ™om˜in—tori—l o˜ je™ts vi— ™om˜iE

n—tori—l to ols su™h —s ˜ije™tionsF

‡e do not oer — sp e™i™ form—liz—tion of the notion of ’˜ije™tive pro of 4 in

this p—p erF ‚—ther we oerD —nd exploreD — distin™tion ˜ etween we—k ˜ije™E

tive pro ofs —nd strong ˜ije™tive pro ofs th—t should ˜ e relev—nt in virtu—lly

—ny p—rti™ul—r form—l ™ontextF vet us now expl—in this distin™tionF e ™om˜iE

n—tori—l identity ™on™erns two indexed f—milies of nite setsD s—y fe g —nd

i iPs

ff g D —nd —sserts th—t for —ll iD je j a jf jF por our purp oses nothing will

i iPs i i

˜ e lost if we sp e—k —s though i were xedD so here—fter we drop the su˜s™riptF

xowD let @
A ˜ e —n —r˜itr—ry ™onstru™tion th—t pro du™es new nite sets from

old su™h th—t the ™—rdin—lity of ƒ determines the ™—rdin—lity of ƒ for —ll ƒ F

„hen — m—nifest ˜ije™tion ˜ etween e —nd f —lso proves th—t jej a jf jY

we ™onsider this — we—k ˜ije™tive proof th—t jej a jf jD in—smu™h —s the f—™t

th—t the ™—rdin—lity of ƒ determines the ™—rdin—lity of ƒ is not itself proved

˜ije™tivelyF e strong ˜ije™tive proof th—t jej a jf j must exhi˜it — m—nifest

˜ije™tion ˜ etween the sets e —nd f themselvesF

es we sh—ll seeD for some ™onstru™tions @
A it is —lw—ys p ossi˜le to dene —

˜ije™tion f X e 3 f just in terms of —ny given ˜ije™tion p X e 3 f D —nd I

not t—king into —™™ount of the sp e™i—l n—ture of the elements of the sets e —nd

f F €rovided th—t the denition of f from p is within one9s form—l me—nsD

one is then gu—r—nteed th—t — we—k ˜ije™tive proof using @
A determines —

strong ˜ije™tive proofF por our purp osesD to s—y th—t the denition of f from

p is within one9s ’form—l me—ns4 is to s—y th—t f ™—n ˜ e ™omputed from p D

in — s™en—rio th—t we will m—ke pre™ise for the s—ke of deniteness @though

the theorems th—t we prove —re not overly sensitive to the pre™ise det—ils

of the mo delAF ‡hen the n—ture of @
A p ermits us to ™ompute f from p D

we s—y th—t su™h — @
A ™—n ˜ e ee™tively ™—n™eledF ‡e give ex—mples of

™onstru™tions @
A th—t ™—n ˜ e ee™tively ™—n™eledD —long with others th—t

™—nnotF

wore gener—llyD supp ose th—t @
Y
A is — ™onstru™tion th—t t—kes two —rguE

ments —nd th—t g is some sort of stru™tured setD — notion th—t we sh—ll m—ke

pre™ise l—terF ƒupp ose — ˜ije™tion p X @eY g A 3 @f Y g A is givenF ‡e

derive — ™riterion for de™iding whether — ˜ije™tion f X e 3 f ™—n ˜ e dened

solely in terms of p —nd the given stru™ture on g F

„his ™riterion is —lge˜r—i™ in n—tureD —nd in sp e™i™ —ppli™—tions it devolves

into questions —˜ out p ermut—tion represent—tions of nite groupsF ‡hen

the ™riterion is s—tisedD — ™—n™ell—tion prin™iple is gu—r—nteed to existD —nd

we m—y pro ™eed to seek it outF sndeedD we h—ve typi™—lly found th—t when

the ™—n™el—˜ility ™riterion is s—tisedD — ™—n™ell—tion pro ™edure exists whose

runningEtime is ˜ ounded ˜y — p olynomi—l fun™tion of the sizes of the sets in

questionF

sf the ™riterion is not s—tisedD then there is no p oint in se—r™hing for — ™—n™elE

l—tion pro ™edureF roweverD even su™h neg—tive results h—ve p ositive imp ort

of — ™ert—in kind for the working ™om˜in—tori—listF por inst—n™eD knowing

th—t the p ower set ™onstru™tion ™—nnot gener—lly ˜ e ™—n™eled suggests the

e f

p ossi˜ility of lo oking for interesting ˜ije™tions ˜ etween P —nd P even in

situ—tions where no interesting ˜ije™tion ˜ etween e —nd f is —v—il—˜leF sn

—ny ™—seD questions —˜ out ™—n™el—˜ilityD when run through the mill of our

gener—lEpurp ose m—™hineryD give rise to interesting questions —˜ out p ermut—E

tion represent—tions of nite groupsD —nd thus ™—n ˜ e — resour™e for —lge˜r—ists

in se—r™h of novel questions to p onderF

yur metho ds ™ome from group theory —nd —re purely m—them—ti™—lD —s opE

p osed to met—m—them—ti™—lF xevertheless it m—y ˜ e fruitful to interpret our P

results —s st—tements —˜ out ™om˜in—tori™s in v—rious top osesD though we do

not pursue thisF

yur work ™—n ˜ e seen —s following in — tr—dition initi—ted ˜y edri—no q—rsi—

—nd ƒtephen wilne ‘R“ in their study of the involution prin™iple —nd ™ontinued

˜y rer˜ ert ‡ilf ‘IP“ —nd f—sil qordon ‘S“ in their work on the prin™iple of

in™lusion —nd ex™lusionF

elso of some relev—n™e is work of v—szlo vov—sz —nd ‚o˜ ert eppleson on

nite sets with rel—tion—l stru™tures @see ‘I“ —nd the referen™es ™ited thereinAF

„heir work diers from ours in th—t we reg—rd two nite stru™tures —s ˜ eing

equiv—lent if they —re den—˜le from one —notherD where—s those —uthors

require th—t the stru™tures —™tu—lly ˜ e isomorphi™F xeverthelessD there —re

m—ny p oints of simil—rity ˜ etween their p—p ers —nd our p—p erD in terms of

˜ oth metho ds used —nd results o˜t—inedD —nd it would ˜ e go o d to underst—nd

˜ etter the rel—tionship ˜ etween the two theoriesF

yur p—p er is org—nized —s followsF sn ƒe™tion P we give prelimin—ries ™onE

™erning group —™tions —nd x some not—tionF xextD in ƒe™tion Q we intro du™e

— proto ™ol to m—ke denite wh—t we me—n ˜y ee™tive ™—n™ell—tionF „hen

we prove „heorem ID the f—si™ iquiv—ri—n™e griterionD to form—lize the intuE

itive notion ’ee™tively ™—n™el—˜le ™om˜in—tori—l ™onstru™tion4 @ƒe™tion RAD

—nd we give sever—l v—ri—tions thereof @ƒe™tion SAF „heorem P re™—sts the

f—si™ iquiv—ri—n™e griterion in — more useful —nd more eleg—nt form @ƒe™E

tion TAF ƒe™tion U digresses to tre—t some issues rel—ted to — v—ri—nt form

of ™—n™ell—tionD for the s—ke of ™ompletenessF ƒe™tion V shows th—t issues of

’uniformity4 of ™—n™ell—tion pro ™eduresD whi™h —re not —ddressed elsewhere

in the —rti™leD ™—n ˜ e given — pre™ise — formul—tion if one mo dies the fr—meE

work so th—t the role pl—yed ˜y —™tions of groups is t—ken over ˜y —™tions

of monoidsF „he ve se™tions th—t follow —pply „heorem P to o˜t—in ™onE

™rete resultsD some p ositive —nd some neg—tiveD some new —nd some oldX —

p ositive result for ™—n™ell—tion of disjoint union with — xed set @oldD ˜ut

we o˜serve how it en™omp—sses the ™ele˜r—ted involution prin™iple of q—rsi—

—nd wilneD there˜y tting th—t te™hnique into our ru˜ri™AY — neg—tive result

for g—rtesi—n pro du™ts @oldD though not previously st—ted in this form in the

liter—tureD —t le—st to our knowledgeAY — p ositive result for g—rtesi—n pro dE

u™ts with p ointed sets @newAY — p ositive result for g—rtesi—n p owers @newAY

—nd — gener—lly neg—tive result for p ower sets @newA whi™h is ™l—ried ˜y the Q

solution of —n interesting groupEtheoreti™—l pro˜lemF ƒe™tion IR gives — sli™k

pro of of — very gener—l equiv—ri—n™e ™riterionF ƒe™tion IS situ—tes our work

in — more —˜str—™tD ™—tegoryEtheoreti™ settingF pin—llyD ƒe™tion IT tre—ts the

relev—n™e of our p—p er to issues in the found—tions of m—them—ti™sF yp en

pro˜lems —re intersp ersed throughout the p—p erF

st is not ne™ess—ry to underst—nd the found—tion—l m—teri—l in the rst h—lf of

the p—p er to p eruse the v—rious expli™it ex—mples in ƒe™tions W through IQY

indeedD these se™tions ™ont—in the m—in fruits of our workD —nd the re—der

m—y wish to s—mple them ˜ efore studying the ro otsD trunkD —nd ˜r—n™hesF es

— middle ™ourseD the re—der might st—rt with ƒe™tions PD QD RD —nd T ˜ efore

pro ™eeding to ƒe™tions W through IQF

ƒome of our su˜sidi—ry prop ositions h—ve trivi—l pro ofs th—t m—y not ˜ e

trivi—l for the re—der to nd un—idedY we h—ve in™luded ™omp—™t versions

of these pro ofsF roweverD r—ther th—n try to follow these veri™—tionsD the

re—der might nd it more helpful to dr—w the —ppropri—te di—gr—ms —nd ™h—se

—rrows in the usu—l w—yD ™onsulting our pro ofs only —s — l—st resortF

PF €relimin—ries

‡e denote the integers ˜y D the n—tur—l num˜ers @in™luding HA ˜y xF ell

other sets ™onsidered —re —ssumed niteF

e qEset is — set with —n —™tion ˜y — group q @on the leftAF @por gener—l

˜—™kground on group —™tions on nite setsD see gh—pter I of ‘T“F sf r is

— su˜group of qD the ™osetEsp—™e qar is —lso — qEsetD under the denition

H H

g @g r A a @g g Ar F e ˜ije™tion f X e 3 f ˜ etween qEsets is — qEset isomorE

phism if g f a f g for —ll g P qF e qEor˜it is — tr—nsitive qEsetF ivery qEset

is — disjoint union of qEor˜itsF „he st—˜ilizerD ƒt—˜ xD of x in — qEset is the

group fg j g x a xgF sf r is — su˜group of ƒt—˜ xD we s—y r st—˜ilizes xF

vet y ˜ e — qEor˜itF por x P y D the m—p from y to qa@ƒt—˜ xA sending

g x to g @ƒt—˜ xA for —ll g P q is — wellEdened qEset isomorphismF sf y

™ont—ins —n element whose st—˜ilizer is r D we s—y th—t y is — qEor˜it of

I

r Etyp eF ƒin™e ƒt—˜ g x a g @ƒt—˜ xAg for —ll g P qD the typ e of —n or˜it is

determined pre™isely up to ™onjug—™yF ƒo we h—ve — ™orresp onden™e ˜ etween

isomorphism ™l—sses of qEor˜its —nd ™onjug—™y ™l—sses of su˜groups of qF R

vet v@qA ˜ e the l—tti™e of su˜groups of qF por —ny qEset ƒ D we now dene

C

m—ps ' Y' Y( from v@qA to xF vet ' @r A ˜ e the num˜er of elements of

ƒ ƒ ƒ

ƒ

C

ƒ whose st—˜ilizer is r F vet ' @r A ˜ e the num˜er of elements of ƒ whose

ƒ

st—˜ilizer ™ont—ins r D iFeFD the num˜er of elements of ƒ th—t —re xed ˜y

r F vet ( @r A ˜ e the num˜er of qEor˜its of ƒ of r Etyp eF i—™h fun™tion is

ƒ

™onst—nt on e—™h ™onjug—™y ™l—ss of su˜groups of qF ivery xEv—lued fun™tion

th—t is ™onst—nt on ™onjug—™y ™l—sses of su˜groups is ( @r A for some qEset

ƒ

ƒ F

C

i—™h of the fun™tions ' @r AD ' @r AD ( @r A determines the othersD —s followsF

ƒ ƒ

ƒ

fy denition

C

@r A a ' ' @u A X

ƒ

ƒ

u r

yn the other h—ndD writing this —s

C

' @u A @r A ' @r A a '

ƒ ƒ

ƒ

u 'r

C

˜y re™ursion down the l—tti™e of su˜groups —llows us to ™ompute ' from '

ƒ

ƒ

@iFeFD ˜y wo˜iusinversionY see ‘T“AF

I

por x in — qEor˜it y D ƒt—˜ g x a ƒt—˜ x i g @ƒt—˜ xAg a ƒt—˜ x or equivE

—lently g P x @ƒt—˜ xAD the norm—lizer of ƒt—˜ x in qF „his shows th—t the

q

num˜er of elements in —n or˜it of r Etyp e with st—˜ilizer r is jx @r Aar jD

q

—nd so

' @r A a jx @r Aar j ( @r A X

ƒ q ƒ

ƒin™e the fun™tion ( @r A is ™le—rly — ™omplete numeri™—l inv—ri—nt for the

ƒ

C

isomorphism typ e of the qEset ƒ D the fun™tions ' @r AD ' @r A —re —s wellF

ƒ

ƒ

i—™h turns out to ˜ e useful in its own w—yF

‡e write fij for the ™—tegory whose o˜ je™ts —re nite sets —nd whose morE

phisms —re ˜ije™tions —nd fij@eY f A for the set of ˜ije™tions from e to f F

„he unde™or—ted expression rom@eY f A st—nds for the set of —ll fun™tions

from sets e to f Y if we me—n — romEset in some other ™—tegory we indi™—te

su™h vi— — su˜s™riptF „he symmetri™ group on the set ƒ is written ƒym ƒ F

por the ˜—si™ vo ™—˜ul—ry of ™—tegory theoryD —nd mu™h moreD see ‘W“F S

QF e™tive —n™ell—tion €ro ™edures

„o m—ke progressD we must —n—lyze the intuitive notions ’purely ™om˜in—toE

ri—l ™onstru™tion4 —nd ’ee™tively ™—n™el—˜le ™om˜in—tori—l ™onstru™tion4 in

se—r™h of —ppropri—te form—l notionsF e ’purely ™om˜in—tori—l ™onstru™tion4

@
A t—kes — given set e —nd pro du™es eD — set whi™h gener—lly ™—rries

some further stru™ture —s wellF xote th—t we need not sp e™ify the ™—tegory

in whi™h @
A t—kes its v—luesF

fije™tions e 3 f indu™e ˜ije™tions e 3 f ˜ e™—use we —re —ssuming

th—t @
A t—kes no —™™ount of the n—mes of p—rti™ul—r elementsF sn f—™tD

@
A determines @˜ut is not identi™—l toA —n endofun™tor  of the ™—tegory

of nite sets —nd ˜ije™tive m—psD iFeF — sp e™ies in the sense of eF toy—l @see

‘U“AF „he usefulness of this not—tion—l distin™tion will ˜ e™ome evident in

ƒe™tion VD where ™om˜in—tori—l ™onstru™tions @
A —re in™—rn—ted —s fun™tors

˜ etween dierent resp e™tive ™—tegoriesF woreoverD we view ™om˜in—tori—l



™onstru™tion —s —n inform—l notionY ™—tegory theory provides one l—ngu—ge

for form—lizing thisD ˜ut we prefer not to prem—turely ™ommit ourselves to —

p—rti™ul—r form—liz—tionD sin™e there m—y ˜ e —sp e™ts of the n—ive notion th—t

—re not ™—ptured ˜y —ny p—rti™ul—r —˜str—™tionF ‡e will ™ontinue to write

e for e when we wish to emph—size —ny extr— stru™ture the set ™—rries

˜y dint of its ™onstru™tion @eFgFD the n—tur—l involution on e a e  eAF

ren™eforth we will —ssume th—t the endofun™tor  is f—ithfulF

‡e give — pre™ise me—ning to ee™tive ™—n™el l—tion in terms of — proto ™ol

involving two p—rtiesD €rogr—mmer —nd w—™hineF @„hese n—mes —re me—nt

to emph—size the rel—tionship ˜ etween the p—rties r—ther th—n their hum—n

or me™h—nisti™ qu—litiesFA

pix — ™om˜in—tori—l ™onstru™tion @
A —nd two disjoint nite sets e —nd f

of ™—rdin—lity nF

essume @for nowA th—t €rogr—mmer —nd w—™hine h—ve —greed on n—mes

for the sets e —nd f themselves @˜ut not for the elements they ™ont—inAF

essume th—t they h—ve —lso —greed on — ™ommon metho d ˜y whi™h to t—ke

— given set of sym˜ols representing the elements of — set ˆ —nd represent

@p ossi˜ly nonEuniquelyA the elements of ˆ —s nite strings whi™h ™onsist

of sym˜ols from — xed nite ™ommon —lph—˜ et together with the sym˜ols T

th—t represent the elements of ˆ F ‡e will here—fter simplify m—tters ˜y

thinking of the elements of ˆ —s ˜ eing themselves the sym˜ols th—t —re used

to represent those elementsD sin™e this is merely — philosophi™—l errorD —nd

not — m—them—ti™—l mist—keF

et the ˜ eginning of the proto ™ol €rogr—mmer knows only the ™—rdin—lity nD

˜ut w—™hine h—s rosters of elements for sets eD f —nd — ˜ije™tion p X e 3

f @given in terms of the strings des™ri˜ ed —˜ oveAF ‡e —ssume w—™hine h—s

un˜ ounded ™omput—tion—l resour™esF sn p—rti™ul—rD w—™hine is —˜le to m—ke

—r˜itr—ry ™hoi™esF

huring the proto ™olD €rogr—mmer sends instru™tions to w—™hine —nd never

re™eives feed˜—™kF €rogr—mmer knows the gener—l w—y in whi™h w—™hine will

present the elements of e —nd f in terms of the elements of e —nd f D

resp e™tivelyD ˜ut l—™king rosters for e —nd f €rogr—mmer ™—n never refer to

the the individu—l mem˜ers of the sets e —nd f F ‡e m—y now m—ke —

henitionX en ee™tive ™—n™el l—tion pro™edure is — xed sequen™e of instru™E

tionsD dep ending only on nD for €rogr—mmer to send to w—™hine whi™h up on

exe™ution —lw—ys ™ulmin—tes in w—™hine9s ™onstru™ting — ˜ije™tion f X e 3 f

whi™h dep ends only on p D not on —r˜itr—ry ™hoi™es m—de ˜y w—™hineF

‡hile we do not stipul—te th—t —n ee™tive ™—n™ell—tion pro ™edure must reE

turn f when given p a f D from —ny given ee™tive ™—n™ell—tion pro ™edure

we e—sily o˜t—in —n ee™tive ™—n™ell—tion pro ™edure th—t do es h—ve this propE

ertyX €rogr—mmer simply —sks w—™hine to return f should p a f @—t most

one su™h f exists ˜ e™—use  is f—ithfulA —nd to follow the given pro ™edure

otherwiseF

RF e —si™ quiv—ri—n™e riterion

en ee™tive ™—n™ell—tion pro ™edure for @
A determines — fun™tion

p X fij@eY f A 3 fij@eY f A X

eYf

ƒin™e €rogr—mmer ™—nnot refer to the n—mes of elements of e —nd f D the

fun™tion p must ˜ e inv—ri—nt under rel—˜ elingsF wore ex—™tlyD the group

eYf U

ƒym e  ƒym f —™ts on fij@eY f A ˜y

I

@& Y& Af a & f &

I P P

I

—nd on fij@eY f A ˜y

I

@& Y& Ap a p & Y

&

I P

P

I

is shorth—nd for &F „his will —lw—ys ˜ e the —™tion th—t we inE where

&

tendD even in the ™—se where e —nd f interse™tF @yne might —t rst think

of intro du™ing — ™omp—ti˜ility ™ondition ˜ etween & —nd & D ˜ut this is in—pE

I P

propri—teY to m—ke sense of the ™on™ept of disjoint unionD for inst—n™eD one

h—s to ™on™eive of there ˜ eing two ™opies of —ny elements in the interse™tionFA

snv—ri—n™e under rel—˜ elings —mounts to

I I

& A a & p @p A& a @& Y& Ap @p A Y p p @@& Y& Ap A a p @

&

P eYf I P eYf eYf I P eYf

P

I

I

th—t isD p must ˜ e @ƒym e  ƒym f AEequiv—ri—ntF

eYf

„he ™ru™i—l ˜ut p erh—ps less o˜vious f—™t is the ™onverseX the mere existen™e

of — @ƒym e  ƒym f AEequiv—ri—nt m—p in rom@fij@eY f AY fij@eY f AA @the

set of —r˜itr—ry set m—ps from fij@eY f A to fij@eY f AA gu—r—ntees —n

ee™tive ™—n™ell—tion pro ™edureF „hough we give — pro of right now in the

spirit of our development so f—rD in ƒe™tion IR ˜ elow the re—der m—y nd —

shortD ™on™eptu—l pro of of — mu™h more gener—l resultF

pirst — trivi—l ˜ut useful

~ ~

AD if p is @ƒym e  A —nd ˜ P fij@f Y vemm— I qiven — P fij@eY

f e

eYf

D given ˜y ƒym f AEequiv—ri—ntD then p

~

~

e

Y

f

I

~ ~ I

@p A a ˜p @ p A Y  A— Y p P fij@ p

f e ˜ —

~

~ eYf

e

Y

f

~ ~

AEequiv—ri—ntF  ƒym is @ƒym

f e

€ro ofX

I I I I I

I I I

I

@ p p p & A a ˜p @ & —A— a ˜p @@˜ & ˜A˜ p —@— & —AA—

& ˜ &

~

~ eYf eYf P

P P

I I I

e

Y

f

I I I I I

I

p @p A& XP AA@— & —A— a & p a ˜@˜ & ˜A@p @

˜ —

~

~ P P eYf

I I

e

Y

f V

rere is the ee™tive ™—n™ell—tion pro ™edureX

pirstD €rogr—mmer instru™ts w—™hine to well{order the set rom@fij@nY nAY

fij@nY nAAD where n a fIY X X X Y ngF sn more det—ilX w—™hine uses the well{

ordering on n to represent e—™h element of n ˜y — unique stringD s—y the

rst lexi™ogr—phi™—lly from —mong the shortest strings representing th—t elE

ement previouslyF ƒin™e e—™h element of n is now represented ˜y — unique

stringD the lexi™ogr—phi™ ordering of these strings determines — well{ordering

of nF w—™hine is now —˜le to represent e—™h ˜ije™tion in fij@nY nA ˜y —

unique string of elements from nX the dom—in of su™h — ˜ije™tion is well{

orderedD so w—™hine ™—n simply list the resp e™tive elements of the r—nge

th—t the elements of the dom—in —re m—pp ed to under the ˜ije™tionF woreE

overD w—™hine ™—n well{order these strings lexi™ogr—phi™—llyD using the well{

ordering on the r—ngeF „his well{orders fij@nY nAF fij@nY nA —nd then

rom@fij@nY nAY fij@nY nAA —re well{ordered simil—rlyF

xextD w—™hine uses the well{ordering on a rom@fij@nY nAY fij@nY nAA

to pi™k the le—st m—p in whi™h is @ƒym n  ƒym nAEequiv—ri—ntY ™—ll it p F

nYn

ƒin™e we —ssume rom@fij@eY f AY fij@eY f AA ™ont—ins — @ƒym e  ƒym f AE

equiv—ri—nt m—p we —re —ssured th—t rom@fij@nY nAY fij@nY nAA ™ont—ins —

@ƒym n  ƒym nAEequiv—ri—nt m—p ˜y vemm— IF

€rogr—mmer now instru™ts w—™hine to ™ho ose @—r˜itr—ry3A ˜ije™tions — X n 3

e —nd ˜ X n 3 f F enother —ppli™—tion of vemm— I yields — @ƒym e 

ƒym f AEequiv—ri—nt m—pD ™—ll it p F pin—lly w—™hine outputs p @p AF

—Y˜ —Y˜

„he ™ru™i—l p oint is th—t the ™hoi™e of — —nd ˜ is imm—teri—lF sf  X n 3 e

—nd X n 3 f —re dierent ™hoi™esD

I I

I I I I

p @p A a ˜p @ @˜ A @ —AA— p A— a ˜p @ p

˜ —

—Y˜ nYn nYn

I I I I

I I

a ˜@˜ A@p @ @p @ AA@ —A— a AA a p @p A X p p

 

nYn nYn Y

‡e h—ve proved the following

ro osition I ije™tions ˜etween @eA —nd @f A ™—n ˜e ee™tively ™—nE

™eled i there exists — @ƒym e  ƒym f AEequiv—ri—nt m—p from fij@eY f A

to fij@eY f AF P W

gomment prom the p oint of view of our motiv—tion it is prop er to sp e—k

of ™—n™eling ™onstru™tionsD not ˜ije™tionsF xeverthelessD when — ™onstru™tion

™—n9t ˜ e ™—n™eled —s — wholeD it is still p ossi˜le th—t p—rti™ul—r ˜ije™tions

p ˜ etween @eA —nd @f A ™—n ˜ e used to dene p—rti™ul—r ˜ije™tions f

˜ etween e —nd f F fy — h—rmless —˜use of l—ngu—geD we s—y th—t su™h —n p

™—n ˜ e ee™tively ™—n™eledF

H H

gomment xote th—t if e —nd f h—ve the s—me ™—rdin—lity —s e —nd f D

H H

then ˜ije™tions ˜ etween e —nd f ™—n ˜ e ee™tively ™—n™eled if —nd only

if ˜ije™tions ˜ etween e —nd f ™—n ˜ e ee™tively ™—n™eledF „h—t isD the

existen™e or nonexisten™e of —n ee™tive ™—n™ell—tion pro ™edure dep ends only

H H

on the ™—rdin—lity n of the sets e —nd f F yn the other h—ndD if e —nd f

h—ve — dierent ™—rdin—lity th—n e —nd f D it ™—n h—pp en th—t ™—n™ell—tion is

p ossi˜le for one p—ir ˜ut not for the otherF

gomment ‡hen — @ƒym e  ƒym f AEequiv—ri—nt m—p from fij@eY f A

to fij@eY f A existsD the pro of of €rop osition I gives —n ee™tive ™—n™ell—tion

pro ™edure whi™h unfortun—tely requires w—™hine to sp end time exp onenti—l

in jejF roweverD —s we h—ve rem—rked e—rlierD in —lmost every ™—se where

we h—ve rst veriedD using groupEtheoreti™ metho dsD th—t the f—si™ iquivE

—ri—n™e griterion is s—tisedD — p olynomi—lEtime ™—n™ell—tion pro ™edure h—s

su˜sequently emergedF

ro lem I o their exist ™—n™el—˜le ™onstru™tions whi™h neverthele ss ™—nE

not ˜e ™—n™eled in time polynomi—l in the ™—rdin—lities of the setsc

ƒome ™om˜in—tori—l ™onstru™tions m—ke use of —n —uxili—ry set g D eFgFD e a

e  g @g—rtesi—n pro du™tAF ƒin™e €rogr—mmer h—s thus f—r ˜ een p ermitted

free referen™e to the m—nner in whi™h elements of ˆ —re sp e™ied in terms

of elements of ˆ D in p—rti™ul—r €rogr—mmer h—s h—d free referen™e ˜y n—me to

the elements of —ny —uxili—ry setF ‡e now seek — renement of €rop osition I

where su™h referen™e is restri™tedF

‡e mo del ™onstru™tions with —uxili—ry sets —s ˜ifun™tors @
Y
AF sn p—rti™uE

l—rD this implies th—t ƒym e —nd ƒym g ˜ oth —™t on @eY g AD —nd th—t these

—™tions ™ommuteF

por the time ˜ eing let us —ssume th—t g is disjoint from e —nd f F IH

‡hen €rogr—mmer is p ermitted no referen™e even to the elements of g D then

—ny ™—n™ell—tion pro ™edure must ˜ e inv—ri—nt under rel—˜ eling the elements of

g —s wellF wore gener—llyD we m—y wish to equip the set g with — ’stru™ture4

—nd then —llow €rogr—mmer referen™e only to D ˜ut not referen™e ˜y n—me

to the elements of g @unlessD of ™ourseD these n—mes —re —n —sp e™t of AF

st turns out th—t our present purp oses do not require us to dene ’stru™ture4

form—lly or to sp e™ify — me™h—nism ˜y whi™h €rogr—mmer will ™ommuni™—te

to w—™hine —˜ out F ell we need —ssume for now is th—t xing su™h —n

determines — su˜group eut g of ƒym g to ˜ e interpreted —s the group of

˜ije™tions whi™h preserve F

„he following result is fund—ment—l to the rest of our workF yne m—y prove

it ˜y mimi™king the pro of of €rop osition IY it is —lso —n immedi—te ™oroll—ry

of the mu™h more gener—l „heorem IQ proved in ƒe™tion IRF

„ eorem I @f—si™ uiv—ri—n™e griterionA ije™tions ˜etween @eY g A

—nd @f Y g A m—y ˜e ee™tively ™—n™eled without referen™e to the elements

of the —uxili—ry set g i there exists — m—p from fij@@eY g AY @f Y g AA to

fij@eY f A whi™h is @ƒym e  ƒym f  eut g AEequiv—ri—ntF P

F t er quiv—ri—n™e riteri—

„heorem IQ in ƒe™tion IR ˜ elow is su˜st—nti—lly more gener—l th—n „heoE

rem I —˜ oveF rere we des™ri˜ e some ™oroll—ries of „heorem IQ th—t pl—y —

supp orting role in our m—in —ppli™—tionsF yther ™oroll—ries ™—n ˜ e found in

ƒe™tion IRF vike „heorem ID e—™h of these results m—y —lso ˜ e proved ˜y

mimi™king the pro of of €rop osition IF

reretofore we h—ve —llowed €rogr—mmer to distinguish ˜ etween the sets e

—nd f F ‡e will sp e—k of —nonymous ee™tive ™—n™el l—tion when referen™e ˜y

n—me to the sets e —nd f themselves is for˜iddenF en —nonymous ee™tive

™—n™ell—tion pro ™edure must ˜ e inv—ri—nt not only under rel—˜ eling the eleE

ments of e —nd f sep—r—telyD ˜ut —lso under the simult—neous rel—˜ eling of

the elements of e —s elements of f —nd vi™eEvers—F

vet ƒym@eY f A ˜ e the su˜group of ƒym @e f A ™onsisting of elements th—t II

resp e™t the p—rtitionF ‡e reg—rd elements of ƒym@eY f A —s p—irs of ˜ije™E

tionsD either of the form @ X e 3 eY X f 3 f A or @ X e 3 f Y X

f 3 eAF „hus ƒym@eY f A is — twoEfold extension of ƒym e  ƒym f F ‡e

extend the @ƒym e  ƒym f  eut g AE—™tion on fij@@eY g AY @f Y g AA to —

@ƒym@eY f A  eut g AE—™tion ˜y setting

I I

I

@ X e 3 f Y  p  X X f 3 eY  Ap a



‡e o˜t—in — ƒym@eY f AE—™tion on fij@eY f A simil—rlyF

I @ uiv—ri—n™e griterion for enon mous e™tive g—n™ell—tionA goroll—r

ije™tions ˜etween @eY g A —nd @f Y g A m—y ˜e ee™tively ™—n™eled withE

out referen™e to e —nd f ˜y n—me i there exists — @ƒym@eY f A  eut g AE

equiv—ri—nt m—p from fij@@eY g AY @f Y g AA to fij@eY f AF

vet  @
Y
A —nd  @
Y
A ˜ e ™om˜in—tori—l ™onstru™tions of two —rgumentsF

I P

gonstru™tions  @
Y
A —nd  @
Y
A —re represented ˜y ˜ifun™tors from the

I P

™—tegory of nite sets —nd ˜ije™tions to itselfD s—y  @
Y
A —nd  @
Y
AF ƒupp ose

I P

th—t jej a jf j i j @eY f Aj a j @eY f AjF „he n—tur—l ƒym e  ƒym f

I P

—™tion on fij@ @eY f AY  @eY f AA —nd previous ™onsider—tions le—d toX

I P

goroll—r P @ uiv—ri—n™e griterion for fifun™torsA ije™tions ˜etween

 @eY f A —nd  @eY f A ™—n ˜e used to ee™tively dene ˜ije™tions ˜etween e

I P

—nd f i there exists — @ƒym eƒym f AEequiv—ri—nt m—p from fij@ @eY f AY  @eY f AA

I P

to fij@eY f AF

por ex—mpleD we might h—ve

@f  f A —nd  @eY f A a @e  f A @f  eAX  @eY f A a @e  eA

P I

en expli™it pro ™edure for ™onstru™ting — ˜ije™tion ˜ etween e —nd f in this

™—se is given in the middle of ƒe™tion IID —fter we h—ve —™quired more to olsF

ƒupp ose th—t ee™tive ™—n™ell—tion is imp ossi˜le for @
Y
A —pplied to eY f

—nd g with — given set of restri™tions on €rogr—mmerF row might we me—E

sure the extent of the f—ilurec yne w—y is to —sk whi™h ˜ije™tions on the

™onstru™ted o˜ je™ts exhi˜it su ™ient l—™k of symmetry th—t they —llow one IP

to dene ˜ije™tions ˜ etween the origin—l o˜ je™tsF @‡e —pply this metho d in

our —n—lysis of the p owerEset ™onstru™tion in ƒe™tion IQFA e dierent —pE

pro—™h is to lo ok for w—ys to m—ke €rogr—mmer9s jo˜ e—sier ˜y p ermitting

€rogr—mmer to refer to some p—rti™ul—r extr— @individu—l or ™olle™tiveA stru™E

ture on the sets eD f —nd g F

qiving ex—mples of p otenti—lly useful stru™tures is e—syF rere —re two ex—mE

ples of individu—l stru™tureD —nd two of ™olle™tive stru™tureF

IF e distinguished p oint in the set g F

PF gy™li™ orderings on the sets e —nd f F

QF en orient—tion of the form—l simplex with vertexEset e f F

f —nd g F RF e ˜ije™tion ˜ etween sets e

@yf ™ourseD we h—ve —lre—dy ™onsidered the utility of referen™e ˜y n—me to

the elements of g —nd the utility of referen™e ˜y n—me to the sets e —nd f D

or put dierentlyD — distinguished element in the set feY f gFA

por —n —ppro—™h to form—lizing the intuitive notion ’stru™tureD4 see ƒe™E

tion IRF por nowD let us —ssume only th—t — ’stru™ture4 on eD f —nd g deE

terminesD if nothing elseD —n —utomorphism group @ƒym@eY f A  ƒym g A

ƒ

whose elements —re the m—ps whi™h preserve F purthermoreD we —ssume h—vE

ing the —˜ility to ™ommuni™—te —˜ out —llows €rogr—mmer to instru™t w—E

™hine to ™onstru™t @eFgFD ˜y ™he™king e—™h element of @ƒym@eY f A  ƒym g A

ƒ

AF en imp ort—nt sp e™i—l ™—se is when is — distinguished for preserv—tion of

element in the set feY f g @whi™h —llows €rogr—mmer to refer e —nd f ˜y

n—meAF „hen is ƒym e  ƒym f  ƒym g F

ƒ

Q @ uiv—ri—n™e griterion it golle™tive ƒtru™tureA ije™tions goroll—r

˜etween @eY g A —nd @f Y g A m—y ˜e ee™tively ™—n™eled m—king use of —

stru™ture on eD f —nd g i there exists — m—p from fij@@eY g AY @f Y g AA

Eequiv—ri—ntF to fij@eY f A whi™h is

ƒ

for purp oses of ™—n™ell—tion dep ends xote th—t the utility of — stru™ture

F sn p—rti™ul—rD — ˜ije™tion p in fij@@eY g AY @f Y g AA ™—n ˜ e only on

ƒ

™—n™eled with referen™e to the stru™ture if —nd only if the interse™tion of

ƒt—˜ p with @viewed —s su˜groups of ƒym@eY f A  ƒ y mg A is ™ont—ined

ƒ

in the st—˜ilizer of some ƒt—˜ f F IQ

‡e ™—n think of the elements of @ƒym@eY f A  ƒym g A th—t preserve some

element of a fij@@eY g AY @f Y g AAD ˜ut preserve no element of fij@eY f AD

—s ™onstituting o˜stru™tions to the ™onstru™tion of — ˜ije™tion ˜ etween e —nd

f F „he set of o˜stru™tions is — ™omprehensive me—sure of the di ™ulty

of ee™tively ™—n™eling @
Y
A —pplied to eY f —nd g F eltern—tivelyD one ™—n

of su˜groups fo ™us on the tot—lity of @ƒym@eY f A  ƒym g A th—t ™ont—in



no o˜stru™tionsD iFeFD th—t h—ve the prop erty th—t ™ont—ins — Eequiv—ri—nt

m—pF „he set of o˜stru™tions determines the ™olle™tion D —nd vi™e vers—D ˜ut



philosophi™—lly it m—y ˜ e helpful to fo ™us on F sn p—rti™ul—rD the m—xim—l



su˜groups in ™orresp ond to the minim—l —mounts of extr— stru™ture th—t



su ™e to f—™ilit—te ™—n™ell—tionF

ƒimil—r me—sures m—y ˜ e developed rel—tive to the presen™e of some given

stru™ture F ‡hen the —v—il—˜ility of do es not su ™e for ee™tively ™—n™elE

@ A to ˜ e the set of su˜groups ing @
Y
A —pplied to eY f —nd g D dene



su™h th—t @ of ™ont—ins — Eequiv—ri—nt m—pF „hen A me—sures the

ƒ 

—mount of further stru™ture needed in —ddition to to ee™t ™—n™ell—tionF

@ ro lem P evelop — theory of the ™ol le™tions A th—t ™—n —rise when



™—n™el l—tion f—ilsF n p—rti™ul—rD ™—n they ˜e expli™itly des™ri˜ed in the ™—se

when  is the power set fun™torc

‡henever eD f —nd g —re not @—s previously —ssumedA p—irwise disjointD

o˜t—in —n equiv—lent ™—n™ell—tion pro˜lem —s followsX m—ke disjoint ™opies of

the setsD ˜ut t—ke —s extr— stru™ture the rel—tions ˜ etween the disjoint sets

indu™ed ˜y the identity rel—tions ˜ etween the origin—l setsF ixtr— stru™ture

™—n only help us ™—n™elD so the —ssumption th—t our origin—l sets were disjoint

™—nnot m—ke ™—n™ell—tion e—sierF ‡e m—ke use of this l—ter when we give

™ounterex—mples to the existen™e of ee™tive ™—n™ell—tion pro ™edures in whi™h

eD f —nd g —re not p—irwise disjointF ƒu™h ex—mples —re often e—sier to think

—˜ outF

F —in eorem

‡ith „heorem P ˜ elowD the question of the existen™e of —n ee™tive ™—n™ell—E

tion pro ™edure for — ™onstru™tion @
Y
A involving — stru™tured —uxili—ry set

g is redu™ed to group theoryF IR

vemm— P et q ˜e — nite group —nd ƒ —nd ˜e qEsetsF hen the

fol lowing —re equiv—lentX

i here is — qEequiv—ri—nt m—p from ƒ to F

ii or every s P ƒ there is — t P su™h th—t ƒt—˜ s ƒt—˜ tF

ro of

@iA @iiAX ƒt—˜ s ƒt—˜ @sA sin™e g s a s implies g @sA a @g sA a @sAF

@iiA @iAX €i™k — represent—tive s of e—™h or˜it y of ƒ F ƒet @s A a t for

y y y

some element t of s—tisfying ƒt—˜ s ƒt—˜ t F ixtend to — m—p from

y y y

ƒ to ˜y sending @g s A to g t F „o see th—t is well{denedD supp ose

y y

I

g s a g s F „hen g g P ƒt—˜ s ƒt—˜ t D so th—t g t a g t —s

I y P y I y y I y P y

P

H H

desiredF „o see th—t is equiv—ri—ntD note th—t for —ll g P qD @g @g s AA a

y

H H H

@@g g As A a g g t a g @ @g s AAF P

y y y

‚eturning now to the f—si™ iquiv—ri—n™e griterionD we now see th—t there is

—n equiv—ri—nt m—p

p X fij@@eY g AY @f Y g AA 3 fij@eY f A

eYf

i for every p P fij@@eY g AY @f Y g AAD ƒt—˜ p is ™ont—ined in one of the

groups ƒt—˜ f with f P fij@eY f AF

sn the spirit of the previous rem—rkD let us s—y th—t — p—rti™ul—r ˜ije™tion

p X @eY g A 3 @f Y g A is ee™tively ™—n™el—˜le if ƒt—˜ p is ™ont—ined in one

of the groups ƒt—˜ f with f P fij@eY f AF „hus the ™onstru™tion @
Y
A is

ee™tively ™—n™el—˜le i every p X @eY g A 3 @f Y g A is ee™tively ™—n™eE

l—˜leF sssues of ’uniformity4 only pl—y — role if we m—ke further restri™tions

su™h —s resour™e ˜ ounds on w—™hineF ƒee —lso ƒe™tion V ˜ elowF

i—™h of the sets eD f D @eY g AD @f Y g AD fij@eY f A —nd fij@@eY g AY @f Y g AA

™—rries — @ƒt—˜ p AE—™tion o˜t—ined ˜y restri™ting the n—tur—l @ƒym eƒym f 

ƒym g AE—™tionF ƒp e™i™—llyX for — P eD @& Y& Y  A@—A a & @—AY for ˜ P f D

I P I

P @eY g AD @& Y& Y  A@ A a AY for P  @ @& Y& Y  A@˜A a & @˜AY for

— — & — ˜

I P I P P

I

I

Y @f Y g A @& Y& Y  A@ A a AY for f P fij@eY f AD @& Y& Y  A@f A a & f &  @

˜ & ˜

I P I P P

P

I

I

I

—nd for p P fij@@eY g AY @f Y g AAD @& Y& Y  Ap a   F p

& &

I P

P I

sn p—rti™ul—r we h—ve IS

vemm— Q he fol lowing two st—tements —re equiv—lentX

i he ˜ije™tion f X e 3 f s—tises ƒt—˜ p t—˜ f F

ii he ˜ije™tion f X e 3 f is —n isomorphism of @ƒt—˜ p AEsetsF

I

ro of ƒt—tement @iA s—ys f a & f & for —ll @& Y& Y  A P ƒt—˜ p Y st—tement

P I P

I

@iiA s—ys f & a & f for —ll @& Y& Y  A P ƒt—˜ p F P

I P I P

eny ˜ije™tion p X @eY g A 3 @f Y g A is itself —n isomorphism vemm—

of @ƒt—˜ p AEsetsF

I I I

I I

 a @& Y& Y  Ap a p D ro of ƒin™e  & p

&

I

I P P

p @@& Y& Y  A@ AA a p A a AA a @& Y& Y  A@p @ AA XP  @  @p @

— & — & — —

I P I P

I P

qiven — group q —nd qEsets e —nd g D @eY g A is — qEset in — n—tur—l w—yD

sin™e  is — ˜ifun™torF ƒupp ose g ™—rries — stru™ture F ‚e™—ll th—t eut g is

the su˜group of ƒym g ™onsisting of ˜ije™tions whi™h preserve F e qE—™tion

on g is E™omp—ti˜le if it is indu™ed ˜y — homomorphism q 3 eut g F

‡e ™—n now st—te @—nd proveA the m—in result of this se™tionD whi™h is used

in ne—rly everything th—t followsF

eorem P uppose the nite set g ™—rries — stru™ture F hen the fol lowE „

ing —re equiv—lentX

or —l l nite groups qD —l l nite qEsets e —nd f D —nd for —l l E

™omp—ti˜le qE—™tions on g D @eY g A —nd @f Y g A —re isomorphi™ if —nd only

if e —nd f —re isomorphi™Y

he ™onstru™tion @
Y
A ™—n ˜e ee™tively ™—n™eled using referen™e

to the stru™ture on g F

ro of @sA @s sAX

vet q a ƒym e  ƒym f  eut g —nd x — ˜ije™tion p X @eY g A 3 @f Y g AF

fy vemm— RD p is —n isomorphism of @ƒt—˜ p AEsetsF es su™hD @sA gu—r—ntees

the existen™e of some @ƒt—˜ p AEisomorphism f X e 3 f F „hen vemm— Q

gu—r—ntees th—t ƒt—˜ p ƒt—˜ f F xowD letting p v—ryD we see th—t @iiA of

vemm— P is s—tisedD so there exists — @ƒym e  ƒym f  eut g AEequiv—ri—nt IT

m—p from fij@@eY g AY @f Y g AA to fij@eY f AF fy the f—si™ iquiv—ri—n™e

griterionD su™h — m—p gu—r—ntees us —n ee™tive ™—n™ell—tion pro ™edure for

 using referen™e to the stru™ture on g F

@s sA @sAX

ƒupp ose @sA f—ilsF „hen we m—y x — nite group qD nonEisomorphi™ qEsets

e —nd f —nd — homomorphism q 3 eut g su™h su™h th—t @eY g A —nd

@f Y g A —re isomorphi™ qEsetsF pix — qEset isomorphism p X @eY g A 3

@f Y g AF x—tur—lly the im—ge of q in ƒym e  ƒym f  eut g is — su˜group

of ƒt—˜ p F ƒin™e e —nd f —re nonEisomorphi™ —s qEsetsD they —re — fortiori

nonEisomorphi™ —s @ƒt—˜ p AEsetsF ƒin™e no m—p f X e 3 f is — @ƒt—˜ p AE

isomorphismD vemm— Q gu—r—ntees th—t there ™—n ˜ e no f su™h th—t ƒt—˜ p

ƒt—˜ f F vemm— P thus pre™ludes the existen™e of — @ƒym e  ƒym f 

eut g AEequiv—ri—nt m—p from fij@@eY g AY @f Y g AA to fij@eY f AD —nd so

there ™—n ˜ e no ee™tive ™—n™ell—tion pro ™edureD even using the stru™ture

on g F P

F non mous —n™ell—tion

„heorem P gives — groupEtheoreti™ ™ondition for ee™tive ™—n™ell—tion when

referen™e ˜y n—me to the sets e —nd f is p ermittedF e further groupEtheoreti™

™ondition must ˜ e met if the ™onstru™tion @
Y
A is to ˜ e —nonymously ee™E

tively ™—n™eled using referen™e to the stru™ture on g F

‚e™—ll the denition of —nonymous ee™tive ™—n™ell—tion from ƒe™tion SF ‚eE

™—ll —lso th—t we s—y th—t — qE—™tion on g is E™omp—ti˜le if it is indu™ed ˜y

— homomorphism q 3 eut g F

eorem Q essume @
Y
A ™—n ˜e ee™tively ™—n™eled using stru™ture on „

g F hen the fol lowing —re equiv—lentX

or —l l nite groups qD nite qEsets eD E™omp—ti˜le qE—™tions on

g D —nd qEset —utomorphisms p of @eY g AD —nd for —l l @Y Y g A P ƒym e 

g a p ƒym e  qD if g p then there exists — qEset —utomorphism f of e



g f F su™h th—t g a f

he ™onstru™tion @
Y
A ™—n ˜e —nonymously ee™tively ™—n™eled

with referen™e to the stru™ture on g F IU

xote th—t the ˜ije™tions  —nd —re not presumed to ˜ e qEset isomorphismsF

ro of @sA @s sAX

ƒupp ose @s sA f—ilsF fy the iquiv—ri—n™e griterion for enonymous ie™tive

g—n™ell—tionD there must ˜ e — ˜ije™tion p X @eY g A 3 @f Y g A su™h th—t

ƒt—˜ p @ @ƒym@eY f A  eut g AA is not ™ont—ined in ƒt—˜ f for —ny f X

e 3 f F xow set q a ƒt—˜ p @ƒym e  ƒym f  eut g A —nd x —n

element @ X e 3 f Y X f 3 eY g A P @ƒt—˜ p A qF ƒin™e @Y Y g A xes

I I

I

g p g g a p p D a p D th—t isD g p F fut @Y Y g A gener—tes ƒt—˜ p

 

over qD so @Y Y g A ™—nnot x —ny f X e 3 f st—˜ilized ˜y qF „h—t isD if f

I I I

is — qEset isomorphismD then g f g a f or equiv—lently g a f g f F

fy the previous „heoremD together with our —ssumption th—t @
Y
A ™—n ˜ e

ee™tively ™—n™eledD q do es x some f X e 3 f F sdentifying e —nd f —long

H

f ™ontr—di™ts @sAF

H

@sAX @s sA

ƒupp ose @sA f—ilsF „hen there is — nite group qD — nite qEset eD —n E

™omp—ti˜le qE—™tion on g D — qEset —utomorphism p of @eY g A —nd —n elE

ement @Y Y g A P ƒym e  ƒym e  q su™h th—t g p D ˜ut for no g a p



qEset —utomorphism f of e do we h—ve g a f g f F vet f ˜ e — qEset isoE

morphi™ to e —nd i X e 3 f ˜ e — qEset isomorphismF @st is helpful to think

of e —nd f —s ˜ eing two ™opies of the ’s—me4 setD —nd i —s the m—p th—t

p X @eY g A 3 @f Y g A is identies the two ™opiesFA „hen the ˜ije™tion

i

I

st—˜ilized ˜y @iY i Y g A P ƒym@eY f A  eut g D ˜ut no qEset isomorphism

I

if X e 3 f @they —re —ll of this formA is st—˜ilized ˜y @iY i Y g AF „hus

ƒt—˜ if D so ˜y vemm— P there ™—n ˜ e no @ƒym@eY f A  eut g AE p ƒt—˜

i

equiv—ri—nt m—pD —nd no —nonymous ee™tive ™—n™ell—tion pro ™edureF P

ro lem Q ind methods for ™he™king the groupEtheoreti™ ™ondition for —nonyE

mous ee™tive ™—n™el l—tionF

F ni orm —n™ell—tion —nd onoids

„his se™tion develops ™riteri— for — more stringent notion of ™—n™ell—tion

germ—ne only to — sp e™i—l ™l—ss of ™om˜in—tori—l ™onstru™tionsF ƒin™e mu™h

of this se™tion p—r—llels e—rlier p—ss—gesD we will in™lude fewer det—ils th—n

˜ eforeF IV

es we h—ve seenD — ™—n™ell—tion for e—™h individu—l ˜ije™tion p X e 3 f

implies the existen™e of —n ee™tive ™—n™ell—tion for the entire ™onstru™tion

D one —lgorithm whi™h w—™hine m—y —pply uniformly to ™—n™el —ll su™h ˜iE

je™tionsD reg—rdless of the ™—rdin—lities of e —nd f F niformity m—y ˜ e™ome

— nonEtrivi—l issue in — v—riety of w—ysD for ex—mpleD ˜y imp osing resour™e

˜ ounds on w—™hine9s —lgorithmF xeverthelessD we sh—ll hen™eforth use the

term uniform to refer sp e™i™—lly to — — ™ert—in typ e of stru™tur—l ™omp—tiE

˜ility ˜ etween ™—n™ell—tions involving dierent ™—rdin—litiesF „his notion of

uniformity will ˜ e prim— f—™ie indep endent of —lgorithmi™ ™on™ernsF

es we ™onsider this new notion of uniform ™—n™el l—tionD p—r—llels —nd ™onE

tr—sts with our previous development will emergeF wonoids —nd ™—tegories

of Esets will pl—y the role formerly pl—yed ˜y groups —nd ™—tegories of qE

setsF „he —n—logue of „heorem P for uniform ™—n™ell—tion do es not gu—r—ntee

ee™tivityF

‡e ˜ egin ˜y dening uniform ™om˜in—tori—l ™onstru™tionsF por simpli™ityD

we restri™t ourselves here to ™om˜in—tori—l ™onstru™tions @
A th—t t—ke just

H H

— single —rgumentF „hroughout the dis™ussionD eY e Y f —nd f will ˜ e nite

setsF y˜serve th—t sin™e e—™h element of @eA is —n equiv—len™e ™l—ss of

strings of elements from e —nd — xed —lph—˜ etD —n equiv—len™e rel—tion on

e indu™es one on @eAF

vet us s—y th—t — ™om˜in—tori—l ™onstru™tion @
A is uniform if it s—tises the

™onditions

H H

e D then @eA @e AY @iA sf e

@iiA sf e is — set of represent—tives for —n equiv—len™e rel—tion on

HH

e D then @eA is — set of represent—tives for the indu™ed equiv—E

HH

len™e rel—tion on @e AF

e uniform ™om˜in—tori—l ™onstru™tion @
A determines —n endofun™tor X



inƒet 3 inƒetD where inƒet is the ™—tegory of nite sets —nd —r˜itr—ry

extends the endofun™tor  X fij 3 fij th—t set m—psF „his endofun™tor



one —lw—ys h—sF

th

x—m les „he n g—rtesi—n p ower ™onstru™tionD whi™h —sso ™i—tes to — set

e the set of strings of length n of elements from eD determines — fun™tor IW

 X inƒet 3 inƒetD ˜ut the ™onstru™tion vinyrd@
AD whi™h —sso ™i—tes to

— set e the set of line—r orders of eD do es not s—tisfy @iAF

xext we expl—in the notions of ee™tive uniform ™—n™el l—tion pro™edure —nd

then uniform ™—n™el l—tion for — uniform ™om˜in—tori—l ™onstru™tionF pix —

e 3 f F ƒupp ose th—t w—™hine is —™tu—lly givenD not p D ˜ut ˜ije™tion p X

 

H H H H

— ˜ije™tion p X e 3 f D su™h th—t p is ™omp—ti˜le with p in the sense

 

H H H

th—t there exist m—ps — X e 3 e —nd ˜ X f 3 f su™h th—t @ ˜Ap a p @ —AF

 

xote th—t neither w—™hine nor €rogr—mmer is given the m—ps — —nd ˜D or

even the sets e —nd f F €rogr—mmer9s go—l is now to dire™t w—™hine in the

H H H

™onstru™tion of — ˜ije™tion f X e 3 f whi™h is to ˜ e likewise ™omp—ti˜le

H

with some ˜ije™tion f X e 3 f in the sense th—t ˜f a f —F woreover we

imp ose — n—tur—l uniformity requirementX f should dep end only on p D not

H

on the ™hoi™e of p D — or ˜F sf this is p ossi˜le for —ll eD f —nd p we s—y th—t

@
A —dmits —n ee™tive uniform ™—n™el l—tion pro™edureF

‡e indi™—te our interpret—tion of the uniformity requirementF sf the m—ps —

H H

—nd ˜ —˜ ove —re surje™tionsD we m—y reg—rd the sets e —nd f —s ™onsisting of

H H H

multiple n—mes for the elements of e —nd f D the ˜ije™tion p X e 3 f —s

 

— redund—nt represent—tion of the ˜ije™tion p X e 3 f D —nd the ˜ije™tion

 

H H H

f X e 3 f —s — ™orresp ondingly redund—nt represent—tion of — ˜ije™tion

f X e 3 f F sf the m—ps — —nd ˜ —re inje™tionsD one m—y reg—rd the sets e —nd

H H H

f —s p—dded versions of the sets e —nd f D the ˜ije™tion p X e 3 f —s —

 

uous d—t—D represent—tion of the ˜ije™tion p X e 3 f p—dded with sup er

 

H

—nd the ˜ije™tion f X e 3 f —s — ™orresp ondingly p—dded represent—tion of

H H

— ˜ije™tion f X e 3 f F „hus ™—n™ell—tion must resp e™t not just rel—˜ elingD

˜ut ™oll—psing —nd p—dding —s wellF

en ee™tive uniform ™—n™ell—tion pro ™edure for @
A determinesD for —ny p—ir

of nite sets e —nd f D m—ps

p X fij@ eY f A 3 fij@eY f A

 

eYf

s—tisfying —n —n—logue of equiv—ri—n™eX

@BA por —ll — X e 3 e —nd ˜ X f 3 f D ˜p @p A a p @p A—

eYf eYf

˜Ap a p @ —AF whenever @

 

wore gener—llyD the uniformity requirement ent—ils th—t the f—mily of m—ps

s—tisfy the following ™omp—ti˜ility ™onditionX PH

H H

@BBA qiven m—ps — X e 3 eD ˜ X f 3 f —nd ˜ije™tions

H H H H

p X e 3 f D p X e 3 f su™h th—t @ ˜Ap a p @ —AD then

     

H

H H

˜p @p A a p @p A—F

e Yf eYf

F vet us ™—ll su™h — ™omp—ti˜le f—mily of m—ps — uniform ™—n™el l—tion for



‡e emph—size th—t — uniform ™—n™ell—tion need not ˜ e ™omput—˜leF

‡e now intro du™e — new form—lism p—rti™ul—rly suited to the study of uniform

™—n™ell—tionF

„o the fun™tor inƒet 3 inƒet we —sso ™i—te the ™omm— ™—tegory X



@ AF ‡e follow the terminology —nd not—tion of ‘W“D ˜ut our tre—tment

 

eY f Y p D where e —nd is selfE™ont—inedF en o˜ je™t in @ A is — triple

 

f —re o˜ je™ts of inƒet —nd p is — fun™tion p X e 3 f F e morphism

 

H H H

A from in @ e Y f Y p to eY f Y p is — p—ir of inƒet morphisms

 

H H H

—A a @ ˜Ap F —Y ˜ a — X e 3 eY ˜ X f 3 f su™h th—t p @

 

worphisms m—y ˜ e visu—lized —s ™ommut—tive squ—resX

H

p

H H

e f

  3

X

— ˜



p

e f

  3

the full su˜ ™—tegory of @ inƒet ‡e denote ˜y A ™onsisting of o˜ je™ts

 



eY f Y p where p h—pp ens to ˜ e — ˜ije™tionF

A to „he forgetful fun™tor from @ inƒet inƒet  inƒet t—kes A @or

 



eY f Y p to p—irs eY f D —nd morphisms —Y ˜ to —Y ˜ F sn p—rti™uE o˜ je™ts

l—rD the not—tion —Y ˜ do es not ˜y itself determine — morphism in @ A

 

H

inƒet sin™e it sp e™ies neither p nor p F „his —m˜iguity turns out to or



˜ e very ™onvenientF

inƒet 3 inƒetD we o˜t—in ™—tE sn the ™—se of the identity fun™tor sd X

egories @sd sdA —nd inƒet F sn this setting — uniform ™—n™el l—tion is

s

merely — fun™tor p X inƒet 3 inƒet whi™h ™ommutes with the forE

s



getful fun™tors just des™ri˜ edD or using the —m˜iguity just notedD — fun™tor

—Y ˜ a —Y ˜ F whi™h s—tises p PI

sf g is — su˜ ™—tegory of inƒet D we ™—ll — fun™tor p X g 3 inƒet

s



s—tisfying p —Y ˜ a —Y ˜ — gEuniform ™—n™ell—tionF sf g h—pp ens to ˜ e the

full ™—tegory of inƒet determined ˜y — ™l—ss of o˜ je™ts g D we —lso refer to



this sort of fun™tor —s — g Euniform ™—n™ell—tionF

y˜serve th—t — uniform ™—n™ell—tion is determined ˜y how it m—ps o˜ je™ts

of inƒet to o˜ je™ts of inƒet F elsoD for — m—pping p from o˜ je™ts of

s



inƒet to o˜ je™ts of inƒet to give — uniform ™—n™ell—tionD it su ™es for

s



p to ˜ e wellE˜ eh—ved on morphisms in the following senseX given — morphism

H H

—Y ˜ X p 3 p in inƒet D —Y ˜ X p p 3 p p is — morphism in inƒet F

s



pun™tori—lity is then —utom—ti™ ˜ e™—use morphisms ™omp ose —™™ording to

their n—mesD —nd these —re preservedF

@‡e ™—ution the re—der th—t the ™ommut—tive squ—res —˜ ove m—y —lso ˜ e

viewed —s morphisms in — dierent ™—tegoryD this time with the verti™—l —rrows

the o˜ je™ts —nd the di—gr—ms ™omp osed side ˜y sideF ‡e most denitely don9t

—sk for p to ˜ e fun™tori—l in this senseD ˜ut see ƒe™tion ISFA

inƒet @respF inƒet A of the form eY f Y p xote th—t the o˜ je™ts in

s



@respF eY f Y f A —re just the elements of fij@ eY f A @respF fij@eY f AAF

 

woreoverD — uniform ™—n™ell—tion p restri™ted to fij@ eY f A is the f—mili—r

 

m—p p F

eYf

@respF A ˜ e the full su˜ ™—tegory es ˜ eforeD we write n for fIY X X X Y ngF vet

I

fij@ nY nA @respF nY nAAF of fij@ inƒet determined ˜y the set

   

naI

naI



ivery morphism in is isomorphi™ @—s —n o˜ je™t in the ™—tegory inƒet



whose o˜ je™ts —re inƒet Emorphisms —nd whose morphisms —re ™ommutE



ing squ—resA to — morphism in F „hus uniform ™—n™ell—tion p X inƒet 3



inƒet exists if —nd only —n Euniform ™—n™ell—tion existsF

s

por e—™h D supp ose p is —n Euniform ™—n™ell—tionF ‡hen nD p nY nY p

is —n element of the nite set fij@nY nAF „his shows ˜y ™omp—™tness th—t some

™onvergesD in — suit—˜le senseD to —n Euniform ™—n™elE su˜sequen™e of the p

l—tion p F ƒin™e the nite su˜ ™—tegories of —re pre™isely the su˜ ™—tegories

9sD we ™on™lude th—t —n Euniform ™—n™ell—tion exists if —nd only if of the

F „his —lso oEuniform ™—n™ell—tions exist for every nite su˜ ™—tegory o of

follows from the ™omp—™tness theorem of rstEorder logi™F

xext we intro du™e — ™ondition on the fun™tor from whi™h it will follow th—t

 PP

nY nAEuniform ™—n™ell—tions — uniform ™—n™ell—tion exists if —nd only if fij@

 

exist for —ll nF

H H H

vet us s—y th—t — inƒet Eo˜ je™t e Y f Y p ™overs o˜ je™t eY f Y p if there



—re morphisms

hi Yi h Y

e e

H H H

eY f Y p 3 e Y f Y p 3 eY f Y p

with ™omp osition the identity m—p on eY f Y p F

vet us s—y th—t the fun™tor is level—˜ le if for every nite su˜ ™—tegory o of



D there exists n su™h th—t e—™h o˜ je™t of o is ™overed ˜y some o˜ je™t inƒet



in fij@ nY nAF

 

is — level —˜le fun™tor with fij@ nY nAEuniform ™—nE vemm— uppose

  

™el l—tions for —l l nF h—s — uniform ™—n™el l—tionF hen



h—s —n oEuniform ™—n™ell—tion whenever o ro of st is enough to see th—t



is — nite su˜ ™—tegory of F ƒupp ose e—™h o˜ je™t mY mY p in o m—y ˜ e

~

nY nY in fij@ nY nAF „o e—™h o˜ je™t ™overed ˜y — ™orresp onding o˜ je™t

p  

mY mY p in o we must —sso ™i—te —n o˜ je™t mY mY f in inƒet in su™h

s

— w—y th—t if

h—Y˜

mY mY p 3 Y Y p

is — morphism in oD then

h—Y˜

mY mY f 3 Y Y f

is — morphism in inƒet F

s

a i — ‡e —ssem˜le the given d—t— into the following di—gr—mD where —nd

—

P

a i X ˜

˜

I PQ



—

E

n n

i i

P

 

P

—

E

m 

~ ~

p p p p

˜

E

m 

i i

I

 

I



˜

E

n n

„he outer squ—re ™ommutes ˜ e™—use e—™h of the inner squ—res do esF „hus

h—Y˜

~ ~

nY nY 3 nY nY

p p

is — morphism in fij@ nY nAF fy —ssumption there is — fij@ nY nAEuniform

   

™—n™ell—tionD so the outer squ—re of the following di—gr—m ™ommutesX

—

E

n n

i i

P

P

—

E

m

~ ~

f f f f

˜

E

m

i i

I

I

˜

E

n n

~ ~ ~ ~

D woreoverD on setting f a i —nd f a i we see th—t sin™e a

f ˜f — f f

I P

~ ~ ~ ~ ~ ~

we h—ve i i ˜ i a D ˜i i a i D ˜ i a i — —nd

f f f — — f f f

I P I I P P

a f —F P ˜f

H I

x—m les vet us ™—ll — fun™tion X e 3 e —t if j @—Aj is ™onst—nt —s —

inƒet 3 inƒet th—t t—kes —t fun™tions v—ries over eF eny fun™tor X



to —t fun™tions is level—˜leF „o see thisD supp ose fp X m 3 m g is the

inƒet set of morphisms of — nite su˜ ™—tegory o of F vet n ˜ e — ™ommon

 PR

multiple of the the m F es long —s X n 3 m is — —t fun™tion th—t is —

left inverse to —n inje™tion i X m 3 nD it is e—sy to nd — ™over

hi Yi h Y

e

~

m Y m Y p 3 nY nY 3 m Y m Y p X

p

sn p—rti™ul—rD —ny ™onstru™tion of the form @eA a g e is uniform —nd

aI

level—˜leF

xow let pun™ e ˜ e the monoid of —r˜itr—ry selfEm—ps of eF iven though

eY f AD given p P we h—ve no —™tion now of pun™ e  pun™ f on fij@

 

eY f AD we will still write ƒt—˜ p for the monoid ™onsisting of —ll p—irs fij@

 

@— X e 3 eY ˜ X f 3 f A su™h th—t @ ˜Ap a p @ —AF ƒimil—rlyD for f P

 

fij@eY f AD we will now write ƒt—˜ f for the monoid ™onsisting of —ll p—irs

@—Y ˜A su™h th—t ˜f a f —F qiven — ˜ije™tion p X e 3 f D —s ˜ eforeD e —nd

f —re n—tur—lly ƒt—˜ p setsF sn this ™ontext it is immedi—te th—t p itself

f Y eAEuniform is — ƒt—˜ p EisomorphismF st is —lso immedi—te th—t — fij@

 

™—n™ell—tion p exists just in ™—se e—™h ƒt—˜ p is in™luded in some ƒt—˜ f F

eYf

„he rest of the pro of of the following theorem follows the lines of the pro of

of „heorem PX

„ eorem et @
A ˜e — uniformD level—˜ le ™onstru™tionF hen the fol lowE

ing —re equiv—lentX

@eA —nd or —l l nite monoids D —l l nite Esets e —nd f D



@f A —re isomorphi™ if —nd only if e —nd f —re isomorphi™Y



he ™onstru™tion @
A h—s — uniform ™—n™el l—tionF P

xote th—t @s sA do es not tell us th—t @
A h—s —n ee™tive uniform ™—n™ell—tion

pro ™edureF sf @sA is s—tisedD —ll th—t our usu—l metho ds show is th—t there

eY f AE is —n ee™tive pro ™edure whi™hD given e —nd f D pro du™es — fij@

 

uniform ™—n™ell—tionF nfortun—tely this f—lls short of —n ee™tive inƒet E



uniform ™—n™ell—tionF „he p oint is th—t the fij@ eY f AEuniform ™—n™ell—E

 

inƒet Euniform ™—n™ell—tionF por this tion we get m—y not extend to —



re—sonD we —re not in — p osition to repl—™e ’uniform ™—n™ell—tion4 in @s sA

˜y ’ee™tive uniform ™—n™ell—tion pro ™edureF4 st is not ™le—r whether this

di ™ulty is intrinsi™ or simply re e™ts the limit—tions of our —ppro—™hF PS

F —n™ell—tion o is oint nion

sn this —nd the following se™tionsD we t—ke eD f D —nd g to ˜ e nonEempty setsF

„he disjoint union e f of two sets e —nd f ™—n ˜ e mo deled —s @fIg 

eA @fPg  f AD —lthough the det—ils of this or —ny other represent—tion —re

unimp ort—ntF „he key p oints —re th—t e —nd f need not ˜ e distin™tD —nd

th—t it is p ossi˜leD given —n element of the disjoint union of e —nd f D to

re™ognize whether it ™—me from e or from f F „o emph—size the —n—logy

with —rithmeti™ —dditionD we will often write e f —s e f F

g is —t on™e g—n™eling the disjoint union ™onstru™tion @ˆ Y g A a ˆ

str—ightforw—rd —nd imp ort—nt in pr—™ti™eF

„ eorem or —ny xed g D the ™onstru™tion e 3 e g ™—n ˜e ee™tively

™—n™eledF

ro of e˜str—™tlyD the existen™e of — ™—n™ell—tion pro ™edure follows from

„heorem P —nd the simple f—™t th—tD for —ny nite group q —nd qEset g D

two qEsets e —nd f —re isomorphi™ if —nd only if qEsets e g —nd f g

—re isomorphi™F sndeedD the ee™t of the fun™tor @
Y g A on the e—™h of the

C

qEset inv—ri—nts ' D ' D ( @dened in ƒe™tion PA is to —dd — ™onst—nt ve™torD

™le—rly —n inverti˜le op er—tionF @ƒee ‘P“FA epplying „heorem P we o˜t—in the

theoremF P

„he following expli™it ™—n™ell—tion pro ™edure is well{known @though in the

p—st it h—s ˜ een st—ted in the ™ontext of the ™omplement—ry ˜ije™tion prinE

™ipleY see ˜ elowAF

prom — ˜ije™tion p X e g 3 f g D dene — ˜ije™tion

f X e 3 f ˜y setting f @—A equ—l to the rst element ˜ P f o˜t—ined ˜y

iter—ting p on — P eF

vet us ™—ll this the iter—tive pro™edure for ™—n™eling disjoint union —nd write

p a f F @„he only — priori ˜ ound on the required num˜er of iter—tions is

jg jFA „he iter—tive pro ™edure is ™le—rly ee™tive sin™e the ™onstru™tion of f

uses only the inform—tion provided ˜y the ˜ije™tion p D not the prop erties or

n—mes of the elements of eD f or g F PT

pred ‚i™hm—n h—s shown th—t even in thisD the simplest inst—n™e of ™—n™elE

l—tion we ™—n think ofD the iter—tive pro ™edure is not the unique ee™tive

™—n™ell—tion pro ™edureF porD supp ose eD f D —nd g —ll h—ve ™—rdin—lity P or

moreF hene — ™—n™ell—tion pro ™edure @dierent from A —s followsF ƒet

p a p D unless p m—ps just two elements of e to g F sf p m—ps just —

I

—nd — to g D set p @— A a p @— AD p @— A a p @— AD —nd p @—A a p @—A

P I P P I

for — not equ—l to — or — F sn the s—me f—shionD when jej a jf j P —nd

I P

jg j jej PD dene — v—ri—nt ™—n™ell—tion ˜y mo difying on ˜ije™tions p

whi™h m—p —l l ˜ut two elements of e to g F @et the end of this se™tionD we

settle the issue of wh—t h—pp ens when one or more of eD f D g h—ve sm—ll

™—rdin—lityFA

st is evident th—t this ™—n™ell—tion prin™iple is equiv—lent to the ™omplemenE

t—ry ˜ije™tion prin™iple @‘S“D ‘II“AY the only dieren™e is th—t in our prin™ipleD

g —nd f g —re identied with one —notherD the two ™opies of g in e

where—s in qordon9s they —re notF vess trivi—llyD we will now show th—t the

involution prin™iple of q—rsi— —nd wilne is —lso ™—n™ell—tion of disjoint unionD

in yet —nother guiseF

e —nd „he involution prin™iple t—kes —s given two ’signed sets4 e a e

C

f a f f D — signEpreserving ˜ije™tion f X e 3 f D —nd two involutions

C

i X e 3 e —nd i X f 3 f D su™h th—t —ll the xed elements of i —nd i

e f e f

˜ elong to the p ositive sets @e —nd f A —nd i —nd i reverse the sign of

C C e f

elements they do not xF „he pro ™edure of q—rsi— —nd wilne then nds —

˜ije™tion ˜ etween the xed p oint sets e —nd f of the two involutionsF

ell we need do is swit™h the neg—tive p—rts of the two signed setsF sn more

~

X det—ilD form sets a e f —nd a f e F hene — ˜ije™tion 3

f

C C

I

~ ~

˜ e the set of p ositive p oints j j ˜y a f j —nd a f j F vet e

f f

e f e f

C

~

not xed ˜y i D —nd dene f simil—rlyF hene — ˜ije™tion X e f 3

f

e

H

C C

~ ~

f e ˜y a i j —nd a i j F j j

f f

e f f f

H H

e e

C

~

xow identify the sets e f —nd f e —long the ˜ije™tion —nd ™—ll

f

H

C C

~

is — ˜ije™tion ˜ etween e the result g F „hen g —nd f g F

f

„he following theorem deline—tes when ee™tive ™—n™ell—tion pro ™edure for

disjoint union is uniqueF „his result will not ˜ e used —g—in —nd m—y ˜ e PU

skipp ed on — rst re—dingF

„ eorem he iter—tive pro™edure is the unique ee™tive ™—n™el l—tion proE

™edure for disjoint union i

PY or jej a jf j

jg j a I —nd jej a jf j RY or

jg j a H —nd jej a jf j a PF

ro of niqueness is trivi—l when jej a jf j a H or IF niqueness is imE

p ossi˜le when jej a jf j a P ˜ e™—use every ee™tive pro ™edure @for —ny

™—n™ell—tion pro˜lemA h—s — distin™t ’opp osite4 o˜t—ined ˜y p ostE™omp osing

P every output f with the nontrivi—l involution on f F ‡hen jej a jf j

—nd jg j PD we expl—ined how to dene — distin™t ™—n™ell—tion pro ™edure in

ƒe™tion WF

„o nish the pro of we need two lemm—sX

uppose th—t —n ee™tive ™—n™el l—tion pro™edure t—kes p X e vemm—

I

g 3 f g to f X e 3 f F uppose moreover th—t je p @f Aj PF hen

I

f @e p @f AA a p @eA f F

I I

ro of pirst note th—t p @e p @f AA a p @eA f D so je p @f Aj a jp @eA

I

f jF „hus is su ™es to prove f @e p @f AA p @eA f F fy hyp othesisD

there —re distin™t elements — P eD i a IY P su™h th—t p @— A a ˜ P f F

i i i

P p @eA f F ‡riting elements of ƒym e —nd ƒym f in ™y™le ƒupp ose f @— A

I

not—tionD we h—ve @@— — AY @˜ ˜ AAp a p D ˜ut @@— — AY @˜ ˜ AAf a f @˜ e™—use

I P I P I P I P

@@@— — AY @˜ ˜ AAf A@— A a f @— AAD viol—ting equiv—ri—n™eF P

I P I P P I

vemm— uppose th—t —n ee™tive ™—n™el l—tion pro™edure t—kes p X e

I

g 3 f g to f X e 3 f F uppose moreover th—t je p @f Aj QF hen

a p j F f j

e f f e

I

I I

ro of vemm— T shows th—t @ p @f AA a e p @f AF vet p A@e

f

I

I

a D —nd supp ose is not the identity m—p on e p @f AF p Aj

f

f e

I

„hen there is some ˜ije™tion & X e 3 e where &@—A a — for — P e p f PV

I

su™h th—t &j @˜ e™—use je p @f Aj QAF do es not ™ommute with

e f

I I I

„hen @&Y p &p Ap a p D ˜ut @&Y p &p Af a f even on e p f D viol—ting

equiv—ri—n™eF P

xow —ll th—t rem—ins is to prove is th—t the ™—n™ell—tion pro ™edure is unique

when jej a jf j Q —nd jg j a H —nd when jej a jf j R —nd jg j a IF ‡hen

jej a jf j Q —nd jg j a HD vemm— U implies f a p F ‡hen jej a jf j R

—nd jg j a ID then f —nd p dier on —t most — single element of eD so there

is no ™hoi™e —˜ out f in this ™—se eitherF P

I F on™—n™ell—tion o €ro du™ts

3 ˆ  g F pix — set xow we ™onsider the g—rtesi—n pro du™t ™onstru™tion ˆ

ƒ D jƒ j ID —nd set ˜ oth e —nd g equ—l to the set of line—r orderings of ƒ F

vet f ˜ e the set of p ermut—tions of ƒ F ‡e m—y dene — ˜ije™tion ˜ etween

e  g —nd f  g ˜y using the f—™t th—t two line—r orderings of ƒ indu™e

— p ermut—tion of ƒ F „here ™—n ˜ e no ™—noni™—l ˜ije™tion ˜ etween e —nd f D

howeverX the set of p ermut—tions of ƒ h—s — distinguished elementD n—mely

the identity p ermut—tionD ˜ut the set of line—r orderings do es notF

wore gener—llyX

eorem or —ny xed g of ™—rdin—lity gre—ter th—n D the ™onstru™tion „

e 3 e  g ™—nnot ˜e ee™tively ™—n™eledF

ro of vet q ˜ e — group of order jg jD whi™h we identify with g F vet q

˜ e the qEset o˜t—ined ˜y h—ving q —™t on itself ˜y tr—nsl—tion @so th—t g

H H

˜ e the qEset o˜t—ined ˜y h—ving q —™t on itself ˜y sends g to g g A —nd q

H H I

™onjug—tion @so th—t g sends g to g g g AF q —nd q ™—nnot ˜ e isomorphi™D

—s q is tr—nsitive —nd q is notF xevertheless X q  q 3 q  q dened

I

@g Y g A a @g g Y g A is —lw—ys —n isomorphism of qEsetsF „he ex—mple ˜y

I P I P

P

in the rst p—r—gr—ph is essenti—lly the sp e™i—l ™—se where q is — symmetri™

groupF @ƒee vemm— I of ‘I“ for — st—tement of wh—t is essenti—lly the s—me

ide—D —pplied in — dierent ™ontextFA P PW

IIF —n™ell—tion o €ointed €ro du™ts

„he situ—tion is —ltogether dierent when we multiply e —nd f ˜y — set

g with — distinguished p ointF es is ™ustom—ryD we ™—ll — set with — disE

tinguished p oint — pointed setY the op er—tion of forming the pro du™t of —

@not ne™ess—rily p ointedA set with — xed p ointed set will ˜ e ™—lled — pointed

produ™t ™onstru™tionF

„ eorem or —ny xed g with — distinguished pointD the ™onstru™tion

e 3 e  g ™—n ˜e ee™tively ™—n™eledF

ro of prom the groupEtheoreti™ viewp ointD the —n—logue of su™h — g is —

qEset with — xed p ointF „husD we must show th—t if e —nd f —re qEsets

—nd g is — qEset with — xed p ointD —nd e  g is isomorphi™ to f  g D then

e is isomorphi™ to f F

C

D the integerE e qEset e is determined up to isomorphism ˜y the inv—ri—ntD '

e

fun™tion on the set of su˜groups of q whi™h re™ords how m—ny elements —re

xed ˜y e—™h su˜group r F „he element @—Y A of eg is xed ˜y r if —nd only

C C C

if ˜ oth — —nd @r A' —re xed ˜y r F „husD for e—™h r D ' @r A a ' @r A a

e e

C C C C

@r A' ' @r A a ' @r AF ƒin™e g ™ont—ins — xed p ointD ' @r A HF st

f e

C C

@r A for —ll r D so th—t e —nd f —re isomorphi™ —s @r A a ' follows th—t '

e e

qEsetsF prom „heorem P it now follows th—t there is — ™—n™ell—tion pro ™edure

for multipli™—tion of sets ˜y the p ointed set g F P

‡e m—y —lso exhi˜it this ™—n™ell—tion prin™iple more expli™itly ˜y the followE

ing ™onstru™tionF e —nd f —re nite setsD —nd g is — nite p ointed set with

— distinguished element ™—lled F vet p X e  g 3 f  g ˜ e — ˜ije™tionF

‡e dene — m—p p X e 3 f ˜y letting p @—A ˜ e the pro je™tion

I I

of p @@—Y AA onto f F e m—p p @a @p A A X f 3 e is dened simil—rlyF

I

p X e f 3 e f pro du™es some ™y™lesF ‡e ster—tion of the m—p p

then use p to p—ir —ny element — of e th—t o ™™urs in — ™y™le with the element

p @—AF „his gives us — nontrivi—l p—rti—l ˜ije™tion ˜ etween e —nd f D s—y from

~ ~

F wultiplying ˜y g we get — nontrivi—l p—rti—l ˜ije™tion from e  g to to

f e

~ ~ ~

A  g to  g F „his indu™es — ˜ije™tion from @e  g to f  g t—king

e f e

~

@f A  g @see ƒe™tion WA —nd now we m—y iter—te the entire pro ™ess until

f

we get — ˜ije™tion from e to f F QH

st should ˜ e noted th—t this ™onstru™tion w—s only dis™overed —fter the groupE

theoreti™—l —ppro—™h suggested its fe—si˜ilityF st is —lso worth verifying th—t

for the ex—mple given in the rst p—r—gr—ph of ƒe™tion IHD if we ™ho ose —

p—rti™ul—r line—r ordering of the set ƒ to serve —s the distinguished p ointD

then the ™onstru™tion we h—ve just des™ri˜ ed indu™es the st—nd—rd ˜ije™tion

˜ etween line—r orderings of ƒ —nd p ermut—tions of ƒ D rel—tive to th—t sp e™ied

line—r orderingF

xoti™e the ™ontr—st with the situ—tion when g h—s no distinguished p ointF

vet q ˜ e eut g D —nd —ssume th—t the —™tion of q on g h—s no xed p ointsF

C C

„hink of the d—t— ' @r A —nd ' @r A @with r v—ryingA —s forming ve™torsD

e e

C

—nd think of the d—t— ' @r A —s forming the entries of — di—gon—l m—trix D

C C C

a ' so th—t ' F ƒin™e the di—gon—l element ' @qA of the di—gon—l

e e

m—trix is zeroD is singul—rD —nd we ™—n nd two qEsets e —nd f su™h

C C

dier ˜y some nonEzero ve™tor in the nullEsp—™e —nd ' th—t the ve™tors '

f e

F „hen e  g —nd f  g —re isomorphi™ —s qEsetsD even though e of

—nd f —re notF „husD if g is some stru™tured setD then g—rtesi—n pro du™t ˜y

g ™—n ˜ e ™—n™eled if —nd only if g h—s — distinguished p oint —s p—rt of its

stru™tureF

ro lem e h—ve just seen th—t if g is — stru™tured nite set —nd e

—nd f —re —r˜itr—ry nite setsD then — ˜ije™tion ˜etween e  g —nd f  g

does not in gener—l yield — ˜ije™tion ˜etween e —nd f unless the stru™ture

on g singles out — p—rti™ul—r pointF oweverD our proof w—s —lge˜r—i™D —nd

not ™om˜in—tori—lF —n one give — more ™om˜in—tori—l underst—nding of this

f—™tc wore spe™i™—l lyD it would ˜e desir—˜le to h—ve some gener—li —tion of

the ™ounterex—mples from e™tion th—t —pplied not just to qEsets g a q

with jg j I on whi™h q —™ts freelyD ˜ut to —l l qEsets th—t h—ve no xed

pointF

e v—ri—nt of the pre™eding pro ™edure for ™—n™eling g—rtesi—n pro du™t ˜y —

p ointed set yields —n expli™it ™—n™ell—tion pro ™edure for the ex—mple menE

tioned in ™onne™tion with the iquiv—ri—n™e griterion for fifun™torsD so we

in™lude it hereF ƒupp ose we h—ve — ˜ije™tion

@f  f A 3 @e  f A @f  eAX p X@e  eA

‡e ™—n re™onstru™t from this — ˜ije™tion ˜ etween e —nd f —s followsF QI

sf e —nd f —re emptyD there is nothing to doF ytherwiseD for

H H

— P eD p @@—Y —AA is either — p—ir @— Y ˜A P e  f or — p—ir @˜Y — A P f  eF iither

w—yD dene — m—p  X e 3 f ˜y @—A a ˜F hene — m—p X f 3 e simil—rlyF

es in the pro ™edure th—t implemented „heorem VD use these two m—ps to

dene — ˜ije™tion ˜ etween two nonEempty su˜sets e —nd f of e —nd f D

H H

resp e™tivelyF ‡rite e a e e D f a f f F „he ˜ije™tion ˜ etween e —nd

H I H I H

f gives us — ˜ije™tion ˜ etween @e  e A @f  f A —nd @e  f A @f  e AF

H H H H H H H H H

st —lso gives — ˜ije™tion ˜ etween @e  e A @e  e A @f  f A @f  f A

H I I H H I I H

—nd @f  e A @e  f A @e  f A @f  e AF ƒu˜tr—™ting these ˜ije™tions

H I I H H I I H

from the ˜ije™tion ˜ etween @e  eA @f  f A —nd @e  f A @f  eA @—g—inD

see ƒe™tion WAD we o˜t—in — ˜ije™tion ˜ etween @e  e A @f  f A —nd

I I I I

@e  f A @f  e AF ‚ep e—t —s neededF

I I I I

st is worth p ointing outD —s —n —sideD th—t the pre™eding pro ™edure ™—nnot ˜ e

gener—lized so —s to —pply to the situ—tion in whi™h one is given — ˜ije™tion

f  —nd e  f  g —nd one wishes to o˜t—in either ˜ etween e  g

— ˜ije™tion ˜ etween e —nd f or — ˜ije™tion ˜ etween g —nd Y forD if one lets

q ˜ e the ™y™li™ group of order T —nd lets eD f D g D ˜ e qEsets whi™h —re

™omp osed of I or˜it of size TD P or˜its of size QD I or˜it of size TD —nd Q or˜it of

size PD resp e™tivelyD then it is e—sy to ™he™k th—t e —nd f —re nonEisomorphi™

—nd g —nd —re nonEisomorphi™ yet e  g f  —nd e  f  g

—re isomorphi™F

enother sort of v—ri—nt of ™—n™ell—tion of p ointed pro du™ts is wh—t one might

™—ll ’rel—tive ™—n™ell—tionF4 qiven — ˜ije™tion ˜ etween e  g  —nd f 

—nd — m—p from g to D one ™—n ™onstru™t — ˜ije™tion ˜ etween e  g g 

—nd f  g F @xote th—t in the ™—se where g h—s — single p ointD so th—t the

m—p from g to is nothing more th—n the ™hoi™e of — distinguished element

of D this ™onstru™tion ™—n™els — ˜ije™tion from e  to f  to o˜t—in —

˜ije™tion from e to f D —s in „heorem VF

qiven —n element of e  g D we ™—n m—p it into e  g 

@using our m—p from g to AD —pply the given ˜ije™tion to get —n element

of f  g  D —nd then pro je™t ˜—™k to f  g F ƒimil—rlyD we ™—n dene —

fun™tion from f  g to e  g F „his gives — p—rti—l ˜ije™tion ˜ etween e  g

—nd f  g F ‡e ™—n lift this to — p—rti—l ˜ije™tion ˜ etween e  g  —nd

th—t —™ts —s the identity on the third ™o ordin—teF g—ll the su˜sets f  g 

of e  g —nd f  g th—t —re in the p—rti—l ˜ije™tion —nd p F „hen our lifted QP

˜ije™tion ˜ etween  —nd p  ™—n ˜ e ’su˜tr—™ted4 from our origin—l

˜ije™tion ˜ etween e  g  —nd f  g  @in — f—shion th—t ˜y now should

˜ e f—mili—rA to yield — ˜ije™tion th—t m—t™hes elements of @e  g  A @  A

with elements of @f  g  A @p  AF xow we —re in — p osition to iter—te

the pro ™edureF

P

xote th—t this ™onstru™tion ™—n ˜ e —pplied in the ™—se where a g D sin™e

we h—ve the di—gon—l em˜edding of g into its g—rtesi—n squ—reF

yne is tempted to link in one9s mind the f—™t th—t je  g j a jf  g j do es not

ne™ess—rily imply jej a jf j @g ™ould ˜ e emptyA —nd the f—™t th—t — ˜ije™tion

˜ etween e  g —nd f  g do es not ne™ess—rily yield — ˜ije™tion ˜ etween e —nd

f @one needs to h—ve — distinguished element of g AF roweverD if the re—der

ex—mines our —rgument ™—refullyD it will ˜ e seen th—t no su™h link ™—n ˜ e found

thereF sn the rst pl—™eD our metho d ™onsiders ™—n™ell—tion pro˜lems ’one

™—rdin—lity —t — time4Y in no ™—se h—ve we —ttempted to des™ri˜ e situ—tions in

whi™h ™—rdin—lities of sets —re unknown to either €rogr—mmer or w—™hineF sn

the se™ond pl—™eD even when g is emptyD our re—son for w—™hine9s in—˜ility

to gener—te — ˜ije™tion ˜ etween e —nd f given — ˜ije™tion ˜ etween e 

g —nd f  g is not th—t the ™—rdin—lities of e —nd f might ˜ e dierent

@they —re known to ˜ e the s—meD ˜y ˜ oth €rogr—mmer —nd w—™hineAD ˜ut

r—ther th—t the ˜ije™tion ˜ etween e  g —nd f  g @empty sets ˜ othA gives

—˜solutely no inform—tionD —nd thus ™—nnot ˜re—k the preEexisting symmetry

of the situ—tionF ƒtillD one ™—n9t help feeling th—t the need for — distinguished

element is in some sense —n ’ee™tive version4 of the hyp othesis th—t g

is nonEemptyD —nd is in™lined to wonder whether some met—Eprin™iple th—t

would expl—in this ™oin™iden™e is w—iting to ˜ e dis™overedF

g—rtesi—n pro du™t ˜y — p ointed set ™—nnot gener—lly ˜ e ™—n™eled —nonyE

mouslyF „o see thisD let e ˜ e — nite su˜set of the re—l line su™h th—t e is symE

metri™—l —˜ out the origin —nd ™ont—ins —n o dd num˜er of p ositive elementsF

vet g ˜ e — nite su˜set of the re—l line su™h th—t g is symmetri™—l —˜ out the

origin —nd do es not ™ont—in HF hene the ˜ije™tion p X e  g 3 e  g ˜y

p @—Y A a @— aj jY AF sf  X e 3 e is multipli™—tion ˜y I —nd X e 3 e

is the identity m—pD then p F fe™—use the p ermut—tion  h—s o dd a p



p—rityD we ™—nnot h—ve  a f f for —ny ˜ije™tion f X e 3 eF ren™eD t—king

q to ˜ e the trivi—l groupD we m—y —pp e—l to „heorem QF

‡e ™on™lude this se™tion with some op en pro˜lemsF QQ

ro lem uppose eY f Y g Y —re nite sets with jej a jf j —nd jg j a

P P

j jD —nd suppose we h—ve — ˜ije™tion ˜etween e R  g  f —nd

P P

Pg P  e  f P  —nd respe™tively denote some xed D where

sets of —nd mutu—l ly distinguished elementsF he qener—l quiv—ri—n™e

riterion of e™tion m—y ˜e used to show th—t we ™—n ™—n™el this ˜ije™tion

to o˜t—in — ˜ije™tion ˜etween e —nd f —nd — ˜ije™tion ˜etween g —nd F s

there — polynomi—lEtime —lgorithm whi™h wil l —™hieve thisc ote th—t the given

P P

˜ije™tion em˜odies the —lge˜r—i™ rel—tion @jej jf jA a P@jg j j jA D —nd th—t

the desired ˜ije™tions em˜ody the rel—tions jej jf j a H —nd jg j j j a HY

™onsequentlyD the soughtEfor ™—n™el l—tion prin™iple ™ould ˜e ™onstrued —s —

™om˜in—tori—l represent—tion of the irr—tion—lity of the squ—re root of F

ro lem f g is — pointed setD does —n inje™tion from e  g to f  g

determine —n inje™tion from e to f c

ro lem f g is — pointed setD does — surje™tion from e  g to f  g

determine — surje™tion from e to f c

—n™ell—tion o €o ers IPF

eorem or —ny positive integer mD the ™onstru™tion e 3 e the mth „

—rtesi—n power of e ™—n ˜e ee™tively ™—n™eledF

ro of vet q ˜ e — nite groupD —nd let e —nd f ˜ e qEsets su™h th—t the

™onstru™ted qEsets e —nd f —re isomorphi™F por —ll su˜groups r of qD

C C C C C C

@' @r AA a ' @r A a ' @r A a @' @r AA D implying ' @r A a ' @r AF st

e e f f e f

follows th—t e —nd f —re isomorphi™ qEsetsF „heorem P then p ermits us to

dedu™e the theoremF P

e ™—n™ell—tion pro ™edure th—t gener—tes — ˜ije™tion in time p olynomi—l in

jej —nd jf j eluded usD ˜ut fortun—telyD e—ron weyerowitz —nd pred ‚i™hm—n

s—w —n e—rly version of this p—p er —nd ™—me up with the followingF

vet p X e 3 f ˜ e — ˜ije™tionD —nd let f X e 3 f ˜ e — ˜ije™tion ˜ etween

H H H

eD f f F @p ossi˜ly emptyA sets e

H H QR

—nd f D so using our proE f indu™es — ˜ije™tion ˜ etween e

H

H H

™edure for ™—n™ell—tion of disjoint unionD we m—y derive from p —nd f —

H

H

˜ije™tion p X e 3 f F ‡e dene — m—p p e f X e e 3 f f

H H

H H

H

˜y letting p @—A ˜ e the rst ™omp onent of the mEtuple p @—Y —Y XXXY —A th—t is

I

not in f F e m—p p X f f 3 e e is dened simil—rlyF €ro ™eeding —s

H H H

I

we did in ƒe™tion IID we m—y use p —nd p to get — ˜ije™tion ˜ etween —

nonEempty su˜set of e e —nd — nonEempty su˜set of f f F „his p ermits

H H

us to extend our p—rti—l ˜ije™tion f to — more in™lusive p—rti—l ˜ije™tionF stE

H

er—ting this pro ™edureD we will eventu—lly —rrive —t — full ˜ije™tion f ˜ etween

e —nd f F

ro lem t is e—sy to see th—t the method of proof of heorem shows

more gener—l ly th—t — ™om˜in—tori—l ™onstru™tion th—t sends e to — disjoint

H

—rtesi—n powers of e where e is just — single point ™—n ˜e union of

ee™tively ™—n™eledF s there — gener—l s™heme for devising polynomi—lEtime

pro™edures th—t implement ™—n™el l—tionc

yne ™onsequen™e of „heorem W is th—t — ˜ije™tion ˜ etween e  g —nd f  g

does determine — ˜ije™tion ˜ etween e —nd f in the ™—se g a e f F porD —

˜ije™tion ˜ etween e  @e f A —nd f  @e f A yields — ˜ije™tion ˜ etween

@e  eA @e  f A —nd @f  eA @f  f AY if we identify e  f with f  e in

the o˜vious w—y —nd —pply ™—n™ell—tion of disjoint unionsD we get — ˜ije™tion

˜ etween e  e —nd f  f D —nd ™—n™ell—tion of p owers gives us — ˜ije™tion

˜ etween e —nd f F sn ™ontr—stD in the ™—se where g is e @or f AD — ˜ije™tion

˜ etween e  g —nd f  g do es not ne™ess—rily give us — ˜ije™tion ˜ etween

e —nd f D —s the ex—mple given —t the ˜ eginning of ƒe™tion IH showsF

ro lem or wh—t nite sets g dened in terms of e —nd f vi— —nd

 is it the ™—se th—t every ˜ije™tion e  g 3 f  g ™—n ˜e ™—n™eledc

e simple ex—mple shows why —nonymous ™—n™ell—tion @—s in goroll—ry I —nd

„heorem QA is gener—lly imp ossi˜le for the g—rtesi—n squ—re ™onstru™tionF

P

vet e a fIY Ig —nd reg—rd e —s — su˜set of the ™omplex pl—neD th—t isD

P P

—sso ™i—te @ F vet p X e 3 e ˜ e multipli™—tion ˜y F Y A with

I I

vet  X e 3 e ˜ e multipli™—tion ˜y I —nd X e 3 e ˜ e the identity

a p p D ˜ut  a f f for —ny ˜ije™tion f X e 3 e @˜y — m—pF „hen

 QS

p—rity —rgumentD —s in ƒe™tion IIAF „hen t—king q to ˜ e the trivi—l groupD

we —pp e—l to „heorem QF woreoverD e m—y ˜ e repl—™ed ˜y —ny nite set

of ™omplex num˜ersD symmetri™—l —˜ out the originD in whi™h the num˜er of

nonEzero elements is ™ongruent to P mo dulo RF

„he ™—n™ell—tion prin™iples th—t h—ve ˜ een illustr—ted in the —rti™le up to this

p oint provide — useful to olEkit for the ™onstru™tion of m—ny sorts of ˜ije™tionsD

in — w—y th—t do es not require one to get one9s h—nds dirty in det—ilsF ‡e

give two ex—mples th—t we nd —musingF

pirstX qiven — ˜ije™tion ˜ etween e  e  g —nd f  f  g D one ™—n ™onstru™t

— ˜ije™tion ˜ etween e  g —nd f  g F porD multiplying the ˜ije™tion ˜y the

P

identity m—p on g D one gets — ˜ije™tion ˜ etween e  e  g  g @e  g A

a

P

@f  g A F xow one ™—n —pply ™—n™ell—tion of g—rtesi—n —nd f  f  g  g

a

squ—resF

P P P

ƒe™ondX qiven ˜ije™tions ˜ etween e 3 f g D f 3 eg D —nd g 3 ef D

one ™—n ™onstru™t ˜ije™tions ˜ etween eD f D —nd g F ‡e h—ve — ˜ije™tion

P P P P P P

˜ etween @e A a e  e  e —nd @f  g A f  g @e  g A  @e  f A

a a a

f  g  e  eF epplying the rel—tive ™—n™ell—tion pro ™edure dis™ussed in

P

ƒe™tion IID one gets — ˜ije™tion ˜ etween e  e —nd f  g  eF €utting

this in — dierent w—yD we get — ˜ije™tion ˜ etween e —nd e  f  g F fut

in ex—™tly the s—me w—y we ™—n get — ˜ije™tion ˜ etween f —nd e  f  g F

„hus we get — ˜ije™tion ˜ etween e —nd f D whi™h yields — ˜ije™tion ˜ etween

e —nd f F e ˜ije™tion ˜ etween f —nd g ™—n ˜ e o˜t—ined in the s—me w—yD

—nd ™omp osing these ˜ije™tions one gets — ™omp—ti˜le ˜ije™tion ˜ etween e

—nd g F

on™—n™ell—tion o onenti—ls IQF

ƒ ƒ

„he p ower set of ƒ will ˜ e denoted P F ‡e show th—t — ˜ije™tion ˜ etween P

ƒ

—nd P do es not gener—lly indu™e — ™—noni™—l ˜ije™tion ˜ etween ƒ —nd ƒ ˜y

I P

nding — nite group q —nd — p—ir of qEsets ƒ —nd ƒ whi™h —re not isomorE

I P

ƒ ƒ

phi™ even though P —nd P —re isomorphi™F „he ™ounterex—mple is then the

ƒ ƒ

isomorphism ˜ etween P —nd P D ™onsidered simply —s — ˜ije™tion ˜ etween

p ower setsF sn this lightD the qE—™tion —pp e—rs —s rel—˜ elingEsymmetriesF sf

there were — ™—noni™—l indu™ed ˜ije™tion ˜ etween ƒ —nd ƒ it would —lso

I P

p ossess the s—me rel—˜ elingEsymmetriesF „his is imp ossi˜leD sin™e q h—s — QT

dierent —™tion on the two setsF

vet y ˜ e — qEor˜it of r Etyp eD with s P y h—ving st—˜ilizer r F €—iring

elements g s of y with ™osets g r of r gives — qEset isomorphism ˜ etween y

—nd qar F sf u is — su˜group of qD the following —re ™le—rly —ll equiv—lentX

H

@iA g s is in the s—me u Eor˜it —s g s

H

@iiA there exists P u su™h th—t g s a g s

H

@iiiA there exists P u su™h th—t g r a g r

H

@ivA there exist P u D P r su™h th—t g a g

H

@vA u g r a u g r F

vet @u Y r A ˜ e the num˜er of u Eor˜its in y F „he equiv—len™e of @iA —nd @vA

implies th—t @u Y r A is —lso the num˜er of dou˜le ™osets of the form u g r F

H H H

por —ny r ™onjug—te to r D y is —lso of r typ eD so @u Y r A a @u Y r AF es

the num˜er of dou˜le ™osets u g r equ—ls the num˜er of dou˜le ™osets r g u

H I I

@u Y r A a @r Y u AF ƒo for —ny u ™onjug—te to @note @r g u A a u g r AD

H

u D we —lso h—ve @u Y r A a @u Y r AF

„he key formul— is

C r Yu r

' @u A a P Y

P

where r r—nges over represent—tives of the ™onjug—™y ™l—sses of su˜groups of

qF „he left side is the num˜er of su˜sets of ƒ whi™h —re st—˜ilized ˜y u F e

su˜set of ƒ is st—˜ilized ˜y u ex—™tly if it is the union of u Eor˜its of ƒ F „he

num˜er of u Eor˜its of ƒ is @r Y u A( @r A sin™e e—™h qEor˜it of r Etyp e

ƒ

r q

@r Y u A u Eor˜itsF de™omp oses into

x—m le I „—ke q a aP  aP D the ulein REgroupF ƒin™e q is e˜ eli—n

the issue of ™onjug—™y of su˜groups is mo otF „he su˜groups of q —re q

D —nd itselfD the three PEelement su˜groups D —nd f gF „—king the

I P

su˜groups in this orderD the m—trix

I I I I I

I P I I P

I I P I P

@r Y u A“ a ‘

I I I P P

I P P P R

is singul—rD with ve™tor ‘PIII I“ in the kernelF „he line—r tr—nsform—tion

@r Y u A“ t—kes the s—me v—lues on ‘P H H H I“ —nd ‘H I I I H“F vet ƒ ˜ e — ‘

I QU

qEset with two or˜its of typ e q —nd one of typ e f gF vet ƒ ˜ e — qEset with

P

ƒ ƒ

D i a IY PY QF „hen P one or˜it of typ e —nd P —re isomorphi™ qEsetsF

i

x—m le P „—ke q a ƒ D the symmetri™ group on three lettersF „he ™onjuE

g—™y ™l—sses of su˜groups of q —re represented q itselfD the QEelement ™y™li™

gener—ted ˜y — re su˜group g D —ny of the PEelement su˜groups e™tionD

i

—nd f gF „—king the su˜groups in this orderD the m—trix

I I I I

I P I P

‘ @r Y u A“ a

I I P Q

I P Q T

is —lso singul—rD with ve™tor ‘P I P I“ in the kernelD so the line—r tr—nsE

form—tion ‘ @r Y u A“ t—kes the s—me v—lues —t ‘P H H I“ —nd ‘H I P H“F @xote

th—t the su˜groupEv—lued indi™es r Y u in the m—trix denoted ˜y ‘ @r Y u A“

r—nge over — system of represent—tives of ™onjug—™y ™l—sses of su˜groupsFA

gF vet ƒ ˜ e vet ƒ ˜ e — qEset with two or˜its of typ e q —nd one of typ e f

P I

ƒ ƒ

F „hen P — qEset with one or˜it of typ e g —nd two of typ e —nd P —re

i

isomorphi™ qEsetsF

„hus we h—veX

e

„ eorem I he ™onstru™tion e 3 P ™—nnot ˜e ee™tively ™—n™eledF P

„he situ—tion illustr—ted ˜y the pre™eding ex—mples is f—r from r—reD —s we

will now showF

‡e th—nk qo etz €feier for — demonstr—tion of the following

„ eorem II he m—trix ‘ @r Y u A“ h—s non ero determin—nt pre™isely when

q is ™y™li™F

‡e h—ve mo died €feier9s —ppro—™h to —void quoting results from represenE

t—tion theoryF pirst we will need —

vemm— et q ˜e — nite group with su˜groups r —nd u F hen the numE

˜er of dou˜le ™osets u g r is the s™—l—r produ™t of the permut—tion ™h—r—™ters

of r —nd u F QV

ro of „he num˜er of dou˜le ™osets u g r is the num˜er of u Eor˜its in qar F

fy the or˜it ™ounting formul—D this is

I I I

jpix j a @ g r a g r A

q r

ju j ju j jr j

Pu Pu Pq

I I

I

@g g P r A a

ju j jr j

Pu Pq

where @ A is I if is true —nd H if notF ƒin™e e—™h p—ir @ Y g A P u  q

I H HI H H

s—tisfying g g P r gives rise to jqj triples @xY y Y A a @g g Y g Y g g A

I I

s—tisfying y xy P u Y P r D we m—y rewrite wh—t we h—d —s x

I I I

I I

@y xy P u Y x P r A

jqj ju j jr j

Pq Pq Pq

I

I I

I I

a @y xy P u A @ x P r A

jqj ju j jr j

Pq Pq Pq

I

I I

@xy u a y u A @x r a r A a

jqj ju j jr j

Pq Pq Pq

I

a jpix xj jpix xj

q u q r

jqj

Pq

—s desiredF P

heory of qroups of inite rder intro du™es — m—trix ‡F furnside9s ™l—ssi™

™—lled there the ’t—˜le of m—rks4 —sso ™i—ted to —ny nite group qF vet

q a f gY q Y X X X Y q a q

I P

˜ e — sequen™e of represent—tives for ™onjug—™y ™l—sses of su˜groups of q

ordered so th—t

jq j


jq j Y jq j

P I

—nd let ˜ e — qEor˜it of q Etyp eD eFgFD qaq F „he ’t—˜le of m—rks4 is then

i i i

the s  s m—trix f a ‘m “ where m is just the num˜er of elements of

i i i

xed ˜y q —re ™onjug—tes of q D so m F „he st—˜ilizers of elements of a H

i i i

unless q is ™ont—ined in some ™onjug—te of q Y in p—rti™ul—r m a H if i F

i i

™ert—inly ™ont—ins —t le—st yn the other h—nd m is —lw—ys p ositive sin™e

i ii

one q Est—˜le elementF „hus ‘m “ is — lower tri—ngul—r m—trix with nonEzero

i i QW

entries on the di—gon—lD —nd the determin—nt of ‘m “ do es not v—nishF ren™e

i

the ™olumns of f —re line—rly indep endentF

xow let g a ‘ “ ˜ e the s  jqj m—trix with rows indexed ˜y the ™onjug—™y

i

™l—sses of su˜groups of qD ™olumns indexed ˜y the elements of qD —nd

i

F „he rows of g equ—l to the num˜er of elements xed when g —™ts on

i

th

—re ˜y denition the p ermut—tions ™h—r—™ters of qF „he n ™olumn of g

™oin™ides with the ™olumn of f ™orresp onding to the @™onjug—™y ™l—ss of A

the su˜group g of q gener—ted ˜y g D sin™e g —nd g le—ve the s—me

elements xedF „he r—nk of g is thus the num˜er of ™onjug—™y ™l—sses of

™y™li™ su˜groups of qF

fy vemm— VD g g is the s  s m—trix a ‘ “D with rows —nd ™olumns

i

indexed ˜y ™onjug—™y ™l—sses of su˜groups of q —nd a jqj @q Y q AF „he

i i

r—nk of ™—nnot ˜ e more th—n the r—nk of g D so for to h—ve full r—nkD

every su˜group of qD in™luding q itselfD must ne™ess—rily ˜ e ™y™li™F

~

a ‘ ~ ™ient for to h—ve full r—nkF vet “ ˜ e „h—t q ˜ e ™y™li™ is —lso su

g

i

the s  s m—trix with rows indexed ˜y the su˜groups of qD ™olumns indexed

˜y — gener—tor for e—™h su˜group —nd ~ equ—l to the num˜er of elements

i

times the num˜er of gener—tors of xed when g —™ts on g F gle—rly

i

~ ~ ~

—mounts to ’™olle™ting4 the identi™—l rows of g F a g g a D sin™e

g g g

~

is inverti˜le sin™e its ™olumns —re nonzero multiples of yn the other h—nd

g

the ™olumns in the t—˜le of m—rksF P

gomment „he previous —rgument gives — formul— for the determin—nt of

a @nA when q a a n X

A a @ A X het@

eltern—tivelyD the determin—nt of — tensor pro du™t is given ˜y the formul—

het@e f A a het@eA
het@f A X

with —nd rel—tively primeD then essenti—lly sf n a

@nA a @ A @ A X

sf is primeD then

I I

het@ @ AA a @ IA
het@ @ AA RH

—nd het@ @ AA a IF

„heorem II gu—r—ntees — ri™h supply of ™ounterex—mples to the ™—n™ell—tion

of the p ower set ™onstru™tionD ˜ut in — sm—ll w—y it h—s — p ositive —sp e™t

—s well it shows th—t — ˜ije™tion ˜ etween p ower sets th—t h—s only ™y™li™

symmetry ™—n in f—™t ˜ e ™—n™eledF

xote th—t the key formul— —˜ ove gener—lizes to

r Yu r C

@u A a ˜ Y for ˜ P '

˜

@where we think of ˜ —s ˜ oth — num˜er —nd —s the ˜Eelement set fHY IY PY XXXY ˜

ƒ ƒ

—nd P —re isomorphi™D then IgAF ƒoD if ƒ —nd ƒ —re qEsets su™h th—t P

I P

ƒ ƒ

˜ —nd ˜ isomorphi™ —s wellD for e—™h ˜ PF „hus even —n innite f—mily

e f

of ˜ije™tions p X ˜ 3 ˜ D ˜ PD m—y still f—il to ee™tively determine —

˜

˜ije™tion ˜ etween e —nd f F

e f

‡h—t is moreD —ny ˜ije™tion p X ˜ 3 ˜ @˜ PA itself ee™tively determines

˜

e f

H H H

H

˜ije™tions p PF sndeedD let q a ƒt—˜ p F „hen even X ˜ 3 ˜ for —ll ˜

˜ ˜

e f

H H

thoughD —s qEsetsD e —nd f m—y not ˜ e isomorphi™D the qEsets ˜ —nd ˜

e f

H H

H

—re isomorphi™D —nd —ny qEset isomorphism p X ˜ 3 ˜ is inv—ri—nt under

˜

rel—˜ elings th—t preserve p F

˜

H

ro lem I s it possi˜le to ee™tively ™onstru™t —n p from p in time

˜ ˜

e He

polynomi—l in m—x@j˜ jY j˜ jAc

‡e —lso —skX

e f

P P e

ro lem II oes — ˜ije™tion ˜etween P —nd P indu™e one ˜etween P

f

—nd P c

e

es — n—l ex—mpleD ™onsider the ’selfEm—p ™onstru™tion4 e a e D the set

of m—ps from e to itselfF ‡e ™l—im th—t this ™onstru™tion ™—nnot ˜ e ™—n™eledF

P Eset with  xed p oints „o des™ri˜ e — ™ounterex—mpleD let us s—y th—t — a

—nd PE™y™les @ P elements in tot—lA is of typ e Y F vet eY f ˜ e of

e

typ e PDI —nd HDPD resp e™tivelyF „hen e @the set of fun™tions from e to eA

f

is isomorphi™ to f D sin™e ˜ oth —re of typ e QPDVF „his ex—mple is typi™—lD

in th—t for m—ny simple ™onstru™tions  for whi™h ™—n™ell—tion prin™iples RI

do not existD one m—y nd ™ounterex—mples using just the PEelement group

q a a P F

e tru™tures rom ld IRF

„he v—ri—tions given in ƒe™tion S seem to p oint the w—y tow—rd in™re—singly

˜—ro que forms of the f—si™ iquiv—ri—n™e griterionF roweverD there is in

f—™t — unied p oint of view whi™h su˜sumes —ll of themD —nd whi™h in some

w—ys is simpler th—n —ny of themF „he go—l of this se™tion is to expl—in this

viewp ointD —nd to provide the setting in whi™h —ll these v—ri—tions ™—n ˜ e

™onveniently provedF

„he key insight is th—t — ˜ije™tion ˜ etween sets e —nd f is itself — kind of

™om˜in—tori—l stru™ture on the set e f X — m—t™hing on the set e f th—t

p—irs elements of e with elements of f F „he situ—tion we —re in ˜ efore we

—re given — sp e™i™ ˜ije™tion ˜ etween e —nd f is —lso — stru™ture on e f D

f into two p—rts of equ—l ™—rdin—lity @e —nd sp e™i™—llyD — splitting of e

f AF por simpli™ityD we will —ssume for now th—t the two ™omp onents of the

p—rtition —re distinguish—˜le from one —notherD though l—ter in this se™tionD

when we develop these ide—s form—llyD we will w—nt to drop this —ssumptionD

sin™e we w—nt our fr—mework to ˜ e —˜le to de—l with —nonymous ™—n™ell—tionF

prom this new p oint of viewD questions —˜ out ™—n™ell—tion ™—n ˜ e phr—sed so

th—t they do not refer to ˜ije™tions —t —llD ˜ut —re merely questions —˜ out the

rel—tive sp e™i—˜ility of ™ert—in kinds of stru™turesF por purp oses of illustr—E

tionD let us t—ke — lo ok —t the the ™—n™ell—tion pro˜lem for g—rtesi—n squ—res

of sets from the new p oint of viewF ‡e —re givenD rst of —llD — splitting of the

P P

set e Y these two splittings —re f D —long with — splitting of the set e f

not unrel—tedD ˜ut must ˜ e ™onson—nt with one —notherD in — w—y th—t will ˜ e

form—lized —s — kind of fun™tori—lityF ‡e —re —lso given — m—t™hing of the set

P P P P

f whi™h is ™onson—nt with the splitting of e f @—g—inD in — sense th—t e

will —mount to nothing more th—n fun™tori—lityD on™e the right denitions —re

f whi™h is ™onson—nt with our in pl—™eAF ‡h—t we seek is — m—t™hing of e

splitting of e f F

f pour kinds of stru™ture pl—y — roleD n—melyD splittings —nd m—t™hings of e RP

P P

—nd e f F st turns out th—t —ll our results t into — ’four stru™tures4

fr—meworkF „hisD in turnD ™—n ˜ e t into — ’two stru™tures4 fr—meworkD where

—ll the given d—t— @initi—lly em˜o died in three stru™turesA ™—n ˜ e rolled up

into — single ™omp osite stru™tureF roweverD ˜ efore we ™—n demonstr—te the

—ppro—™hD we must expl—inD —t l—stD wh—t we me—n ˜y — ’stru™ture4F

hene — nite stru™ture type to ˜ e — p—ir @ƒY p A ™onsisting of — ™—tegory ƒ

—nd — fun™tor p from ƒ to fij s—tisfying

H

—nd ( X 3 is — morphism in fijD then there exists — unique @iA sf p @sA a

o˜ je™t ( s P ƒ —nd — unique ƒEmorphism ' X s 3 ( s su™h th—t p @' Aa( Y

I

@iiA por P fijD p @ A is — nite setF

yne should think of p —s — forgetful fun™tor t—king stru™tured o˜ je™ts to

I

their underlying setsF ‡e s—y th—t p @ A is the set of @ƒY p AEstru™tures on

F ssomorphism of @ƒY p AEstru™tures is simply isomorphism in the ™—tegory ƒF

I I

fy @iA —˜ oveD there is — @ƒym AE—™tion on p @ AF @sndeedD the set p @ A

is —™tu—lly fun™tori—l in FA ‡e dene eut s to ˜ e the su˜group of ƒym p @sA

th—t xes sF

e nite stru™ture typ e is ee™tively presented if in —ddition there is — ™o ding

˜y — @not ne™ess—rily uniqueA string of for —ny @ƒY p AEstru™ture s on —ny

elements of D p ossi˜ly —long with some other supplement—l sym˜ ols from

some nite —lph—˜ etD su™h th—tX

@iiiA —ll the strings th—t represent @ƒY p AEstru™tures on ™—n ˜ e ee™tively

enumer—tedY

@ivA it ™—n ˜ e ee™tively determined whether two strings represent the s—me

@ƒY p AEstru™tureY

H

@vA if ( X 3 is — ˜ije™tionD ƒ is — string representing —n @ƒY p AEstru™ture

H H

—nd the string ƒ is o˜t—ined from ƒ ˜y su˜stituting elements of s on

H

—™™ording to ( D then ƒ represents ( sF for elements of

ell nite stru™ture typ es to ˜ e dis™ussed hen™eforth —re ee™tively presentedF

sf p X ƒ 3 ƒ is — fun™tor su™h th—t p a p p D we s—y th—t the nite

PYI P I P I PYI

stru™ture typ e @ƒ Y p A renes the nite stru™ture typ e @ƒ Y p A vi— p F xote

P P I I PYI

I

th—t eut s —™ts on p @s AF

I I

PYI

henitionX en ee™tive pro™edure for dening — ƒ Estru™ture in terms of —n

P

ƒ Estru™ture s is — xed sequen™e of instru™tions for €rogr—mmer to send to

I I RQ

w—™hineD su™h th—t if these instru™tions —re —pplied ˜y w—™hine to —ny string

th—t ™o des s D the result will ˜ e — string th—t ™o des —n ƒ Estru™ture s D su™h

I P P

th—t s dep ends neither up on whi™h string th—t ™o des s is used ˜y w—™hine

P I

nor up on —ny —r˜itr—ry ™hoi™es th—t m—y ˜ e m—de ˜y w—™hine in the ™ourse

of ™—rrying out €rogr—mmer9s instru™tionsF

vet @ƒ Y p A —nd @ƒ Y p A ˜ e ee™tively presented nite stru™ture typ es su™h

I I P P

th—t @ƒ Y p A renes @ƒ Y p A vi— p F

P P I I PYI

„ eorem IP @ ixe oint griterionA et ˜e — nite setF qiven —n

@ƒ Y p AEstru™ture s on D one m—y ee™tively dene —n @ƒ Y p AEstru™ture s

I I I P P P

on in terms of s su™h th—t p @s A a s if —nd only if the @eut s AE—™tion

I PYI P I I

I

on p @s A h—s — xed pointF

I

PYI

ro of ƒupp ose s s—tisfying p @s A a s is ee™tively dened in terms of

P PYI P I

s D ˜ut with no referen™e to the n—mes of the elements of F „hen for —ny

I

' X 3 whi™h preserves s D we h—ve ' s a s F „hus s is — xed p oint of

I P P P

I

the @eut s AE—™tion on p @s AF

I I

PYI

I

gonverselyD let us —ssume th—t the @eut s AE—™tion on p @s A h—s — xed

I I

PYI

p ointF ƒet n a j j —nd n a fIY PY X X X Y ngF

€rogr—mmer ˜ egins ˜y —sking w—™hine to nd the rst string in the lexi™oE

H

gr—phi™ order whi™h represents —n @ƒ Y p AEstru™ture s on n isomorphi™ to

I I

I

the stru™ture s on F xext progr—mmer —sks w—™hine to ™ho ose — — ˜ije™tion

I

H

a s F  X n 3 su™h th—t s

I

I

xote th—t  gives — wellEordering of F „he resulting lexi™ogr—phi™ ordering

on the set of strings of elements of rst —llows w—™hine to pi™k — unique

I

representing string for e—™h element of p @s AD —nd then indu™es —n order

I

PYI

on the ™hosen stringsF w—™hine m—y there˜y determine — wellEordering of

I

p @s AF

I

PYI

pin—llyD €rogr—mmer —sks w—™hine to output —s s the lexi™ogr—phi™—lly le—st

P

I

@eut s AExed p oint in p @s AF

I I

PYI

‡e must ™he™k th—t s is indep endent of the ™hoi™e of F ƒupp ose w—™hine

P

H H H H I

h—d ™hosen inste—d  X n 3 su™h th—t  s a s F „hen   X 3

I

I

I

H

˜ elongs to eut s F xow the wellEordering of p @s A determined ˜y  m—y

I I

PYI RR

I

˜ e o˜t—ined ˜y pushing forw—rd the wellEordering of p @s A determined

I

PYI

I

H I H I

˜y  —long the —™tion of   on p @s AF ƒin™e   P eut s D the

I I

PYI

I

@eut s AExed p oints in p @s A —re —ssigned the s—me ordin—l ˜y the  —nd

I I

PYI

H

 orderingsF sn p—rti™ul—rD the le—st xed p oints ™oin™ideD so s is wellE

P

denedF P

~ ~

Y p AD @„Y p vet @ƒY p AD @ Y p A —nd @ AD ˜ e ee™tively presented nite stru™E

ƒ „

~ ~

ƒ

ƒ

~ ~

Y p —nd @ Y p A renes @ƒY p A vi— — fun™tor p ture typ esF ƒupp ose th—t @ A

„ ƒ

~ ~ ~

ƒ

ƒ ƒ

Yƒ

renes @„Y p A vi— — fun™tor p F vet  X ƒ 3 „ ˜ e — fun™torF

~

Y

„ eorem IQ @ ener—l uiv—ri—n™e griterionA —ving xed —n @ƒY AE

~

AEstru™ture s~ on Y p stru™ture s on set one ™—n ee™tively dene —n @ su™h

ƒ

~

ƒ

~

~ ~

Y p @s ~A a s from —ny @ th—t p AEstru™ture su™h th—t p @ on A a sD

„ t t

~ ~ ~

ƒ

Y Yƒ

I I

@sAF @sA 3 p if —nd only if there is —n @eut sAEequiv—ri—nt m—p p

~ ~

Yƒ Y

ƒ

ro of ‡e —re going to derive this result from the pixed €oint griterionF

„o —pply the pixed €oint griterion we must sp e™ify — nite set —nd nite

stru™ture typ es @ƒ Y p A —nd @ƒ Y p AD where @ƒ Y p A renes @ƒ Y p A vi— —

I I P P P P I I

fun™tor p F

PYI

„he of the pixed €oint griterion will ™oin™ide with our present F „he

nite stru™ture typ e @ƒ Y p A of the pixed €oint griterion will ˜ e our present

I I

@ƒY p AF yur go—l now is to devise — suit—˜le nite stru™ture typ e @ƒ Y p AF

P P

~

su™h th—t Y p AEstru™ture s~ on „o s—y th—t one ™—n ee™tively dene —n @

ƒ

~

ƒ

~

~ ~

Y p @s ~A a s from —ny @ p AEstru™ture su™h th—t p @ on A a sD is

„ t t

~ ~ ~

ƒ

Yƒ Y

I I

just to s—y th—t one ™—n ee™tively dene — m—p m X p @sAF @sA 3 p

~ ~

Yƒ Y

ƒ

@w—ps m of this sort gener—lize the m—ps p X fij@eY f A 3 fij@eY f A

eYf

we studied e—rlierFA „he ide— is to reg—rd su™h — m—p m —s — stru™ture on

rening sF

xow we —re going to dene the new nite stru™ture typ e @ƒ Y p AF „he ™—teE

P P

gory ƒ will h—ve o˜ je™ts th—t —re triples @mY sY A where is — nite setD s is

P

I I

—n @ƒY p AEstru™ture on —nd m is — m—p m X p @sA 3 p @sAF e morE

ƒ

~ ~

Yƒ Y

ƒ

H H H H

phism X@mY sY A 3 @m Y s Y t A in ƒ is given ˜y — ˜ije™tion ( X t 3 t su™h

P

H H

th—t s a ( s —nd m a ( mF @„he —™tion of ( on m is indu™ed ˜y the —™tions

I I

@sAFA hene the fun™tor p X ƒ 3 fij so th—t of ( on p @sA —nd p

P P

~ ~

Yƒ Y

ƒ

on o˜ je™ts p @mY sY A a —nd on morphisms p @ A a ( F hene the fun™tor

P P RS

p so th—t on o˜ je™ts p @mY sY A a s —nd on morphisms p @ A a ' where

PYI PYI PYI

H

' X s 3 s a ( s is the unique morphism su™h th—t p @' A a ( F

I

I

fy ™onstru™tionD the @eut sAE—™tion on p @sA h—s — xed p oint pre™isely

PYI

I I

when there exists —n @eut sAEequiv—ri—nt m—p from p @sAF „he @sA to p

~ ~

Yƒ Y

ƒ

result now follows from the pixed €oint griterionF P

„wo nite stru™ture typ esD intro du™ed inform—lly e—rlierD will pl—y — distinE

guished role in the study of ˜ije™tionsF „he nite stru™ture typ e ƒ lit will

˜ e the p—ir ™onsisting of

@iA the ™—tegory whose o˜ je™ts —re nite sets e—™h ™—rrying —n

equiv—len™e rel—tion th—t determines ex—™tly two equiv—len™e ™l—ssesD

these of equ—l sizeD —nd whose morphisms —re ˜ije™tions whi™h preE

serve the equiv—len™e rel—tionsY

@iiA the fun™tor from this ™—tegory to fij whi™h forgets the equivE

—len™e rel—tionF

„he nite stru™ture typ e —t™ is the renement of ƒ lit where the o˜ je™ts

—ddition—lly ™—rry — ˜ije™tion ˜ etween the two equiv—len™e ™l—sses whi™h the

morphisms preserveF

„o o˜t—in results —˜ out ™—n™eling ˜ije™tions from the qener—l iquiv—ri—n™e

~

Y p A A to ˜ e renements of ƒ lit —nd @ griterionD just t—ke @ƒY p A —nd @„Y p

ƒ

~

ƒ

ƒ

~

Y p —nd @ A to ˜ e the ™orresp onding renements of —t™ F

„

~

iven with our interests ™entered on ˜ije™tionsD we m—y ˜ e led to ™onsider ™orE

resp onden™es more gener—l th—n ˜ije™tionD —nd so ˜ enet from the exi˜ility

of the qener—l iquiv—ri—n™e griterionF

x—m le fy — signed set we me—n — set e p—rtitioned into — set e of posE

C

itive elements —nd — set e of neg—tive elementsF „o reinfor™e this intuitionD

we will write su™h — signed set —s e e D r—ther th—n —dopting some more

C

—™™ur—te ˜ut unintuitive not—tion like @e Y e AF e should ˜ e thought of —s

C

— ™om˜in—tori—l inst—nti—tion of the integer je j je jD whi™h we denote ˜y

C

jjejjF e signje™tion ˜ etween signed sets e a e e —nd f a f f is

C C

just — ˜ije™tion ˜ etween e f —nd e f F @yne m—y —lso reg—rd — signE

C C

je™tion —s ™onsisting of — signEreversing p—rti—l m—t™hing of the elements of eD

— signEreversing p—rti—l m—t™hing of the elements of f D —nd — signEpreserving RT

˜ije™tion ˜ etween the rem—ining unm—t™hed elements of e —nd of f FA „he

set of signje™tions from e to f we write —s ƒign @eY f AF

„he n—tur—l w—y to multiply — set ˆ ˜y — signed set @™orreE a

C

sp onding to multiplying — n—tur—l num˜er ˜y —n integerD under our interpreE

t—tionA is given ˜y the fun™tor  with @ˆ A a @ˆ  A AF €ut @ˆ 

C

a e —nd a f D with eD f xedF es degener—te ™—sesD we m—y t—ke

C

ˆ a e —nd ˆ a f D resp e™tivelyF st is e—sily proved th—t jjejj a jjf jj

implies jej a jf jY we seek — ™—n™ell—tion prin™iple ™orresp onding to thisF

@ƒin™e — signje™tion ˜ etween e —nd f —mounts to — ˜ije™tion ˜ etween

@e  eA @f  f A —nd @e  f A @f  eAD wh—t we h—ve is just — new

viewp oint on the pro˜lem we p osed in ™onne™tion with the iquiv—ri—n™e griE

terion for fifun™tors —nd solved in ƒe™tion IIFA sn this ™—se the qener—l

iquiv—ri—n™e griterion tells us th—t ee™tive ™—n™ell—tion is p ossi˜le if —nd

only if there is — @ƒym e  ƒym f AEequiv—ri—nt m—p from ƒign@eY f A

~

A in the ™riterion Y p to fij@eY f AF ƒp e™i™—llyD the nite stru™ture typ e @

„

~

would ™onsist of — p—ir of signed sets @pl—ying the roles of e —nd f A —nd

— signje™tion ˜ etween themF

ro lem IP wore gener—l lyD dene @ˆ ˆ A  @ A —s @ˆ 

C C C C

A AF ˆ  @ˆ  ˆ  ow suppose ˆ D D —nd —re signed

C C

setsF qiven — signje™tion ˜etween ˆ —nd with n Q D ™—n one

ee™tively dene — signje™tion ˜etween ˆ   —nd the empty setc

x—m le sn ƒe™tion S we dis™ussed two w—ys of me—suring the f—ilure

when ™—n™ell—tion is imp ossi˜leF eltern—tivelyD when ee™tive ™—n™ell—tion

for @
Y
A —pplied to eY f —nd g f—ilsD we ™ould ™onsider settling for lessF

por ex—mpleD we ™ould seek — nonEempty p—rti—l ˜ije™tion ˜ etween e —nd

f D p erh—ps of — ™ert—in sizeY we ™ould seek some other sort of m—p from e

to f Y we ™ould seek — @sm—llD nonEemptyA set of ˜ije™tionsY we ™ould seek —

e —nd f D where is some other ™onstru™tionF ƒuit—˜ly ˜ije™tion ˜ etween

sp e™i—lizedD the qener—l iquiv—ri—n™e griterion m—y ˜ e ˜rought to ˜ e—r on

—ny of these situ—tionsF ell one need do is sp e™ify suit—˜le nite stru™ture

~

A these —ltern—tive outputsF Y p typ es @

ƒ

~

ƒ

‡e note in p—ssing th—t sm—llD nonEemptyD ee™tively den—˜le sets of ˜iE

I

je™tions —re ™losely rel—ted to sm—ll @eut sAEor˜its in p @sA —s in the pro of

PYI RU

of the qener—l iquiv—ri—n™e griterionD with the qener—l iquiv—ri—n™e griteE

rion ˜ eing the sp e™i—l ™—se of or˜its of size @iFeFD xed p ointsAF „his seeming

gener—liz—tion of the qener—l iquiv—ri—n™e griterion do es not —™tu—lly give

—ny —dded gener—lityD sin™e unions of or˜its in — qEset @of whi™h equiv—ri—nt

sets of ˜ije™tions —re just — sp e™i—l ™—seA —re t—nt—mount to xed p oints in —

derived qEsetD n—melyD the p ower set with the inherited qE—™tionF

I F —te ori™—l ie oint

„ill nowD we h—ve reg—rded ™onstru™tions  —s endofun™tors of the ™—tegory

fij of nite sets —nd ˜ije™tionsF st is ˜ etter here to ™onsider the fun™tor

from fij to fij D the full im—ge of D whi™h is dened —s followsF „he





o˜ je™ts of fij —re the o˜ je™ts of fijD ˜ut



rom @eY f A a fij@eY f A X

fij

is the identity m—p @this —voids the te™hni™—l —nnoy—n™e th—t yn o˜ je™ts



e a f do es not gener—lly imply e a f AF yn m—ps ™oin™ides with F



xote th—t is surje™tive on o˜ je™ts where  is notF



yur notion of — ™—noni™—l ™—n™ell—tion for — ™onstru™tion  lies ˜ etween two

extremesF e ™ert—in nonE™onstru™tive view might reg—rd  —s ™—n™el—˜le

simply if the ™—rdin—lity of e determines the ™—rdin—lity of eF „he purely

fun™tori—l view s—ys  is ™—n™el—˜le if there is — fun™tor from fij ˜—™k to



fij whi™h is — oneEsided inverse to F sn gener—l one ™—nnot exp e™t su™h

fun™tors to existF sndeed when jej jej RD the group homomorphism

@eY eA a fij@eY eA 3 fij@eY eA rom

fij

@eY eA ™onE h—s im—ge of size —t most P sin™e the symmetri™ group rom

fij

t—ins — simple su˜group of index PF sn p—rti™ul—rD the homomorphism ™—nnot

˜ e surje™tiveF

„he question —rises thenD if our equiv—ri—nt m—ps

@eY f A 3 fij@eY f A p X rom

eYf

fij

do not dene fun™torsD how ™lose do they ™omec sndeed we h—ve fun™torElike

g—dgets p D m—pping o˜ je™ts to o˜ je™ts —nd —rrows to —rrows from fij to

 RV

fijF vet us s—y th—t p is — semifun™tori—l ™—n™el l—tion for if for —ny p—ir



of ™omp os—˜le —rrows x —nd y D

p @xy A a p @xAp @y A

F ie™tive provided —t le—st one of x or y is in the im—ge of the fun™tor



™—n™ell—tion pro ™edures give semifun™tori—l ™—n™ell—tionsD ˜ e™—use they do

not dep end on the n—mes of elementsD —nd morphisms in the im—ge of —re



essenti—lly those indu™ed ˜y rel—˜ elingF

xote th—t — single equiv—ri—nt ˜ije™tion p indu™es — semifun™tori—l ™—nE

eYf

™ell—tion on — ™omp onent of the ™—tegoryY there —re no further ™omp—tiE

˜ility ™onstr—intsF sf we t—ke e a f D we —re just lo oking —t m—ps from

rom @eY eA a fij@eY eA to fij@eY eA —s twoEsided fij@eY eAEsetsD so

fij

we —re ˜—™k to group theoryF ƒin™e these —re ˜ oth symmetri™ groupsD we

fr—me the followingX

ro lem IQ l—ssify p—irs @qY r AD with q —nd r symmetri™ groups s—tisE

fying r qD —™™ording to the existen™e of — m—p q 3 r th—t respe™ts q

—nd r —s twoEsided r setsF

purthermoreD we prop oseX

ro lem I xplore the notion of semifun™tori—l ™—n™el l—tion developed in

e™tion for fun™tors outside of ™om˜in—tori™sF

F ound—tion—l sm li™—tions I

sn — series of p—p ers from the IWPH9sD elfred „—rski —nd edolf vinden˜—um

worked on the —rithmeti™ of innite ™—rdin—ls from — p oint of view somewh—t

simil—r to oursY see ‘V“F

ƒome of our theorems m—y ˜ e ™—st —s indep enden™e results —˜ out  D set theE

ory without the exiom of ghoi™eF vet @e A —nd @f A ˜ e two sequen™es

i i

iP iP

of nite sets indexed ˜y the n—tur—l num˜ersD su™h th—t th—t je j a jf j for

i i

every iF ‡e s—y th—t the sequen™es @e A —nd @f A —re isomorphi™ if there

i i RW

is — sequen™e @f A of ˜ije™tions f X e 3 f F ƒu™h isomorphism is —

i i i i

iP

nonEtrivi—l issue —˜sent the exiom of ghoi™eF yur results show th—t if the

P P

sequen™es @e A —nd @f A —re isomorphi™D then so must ˜ e the sequen™es @e A

i

i i

—nd @f AF yn the other h—ndD our results —nd some routine for™ing yield

i

e f

mo dels where @P A —nd @P A —re isomorphi™D ˜ut @e A —nd @f A —re notF

i i

„o e—™h nite stru™ture typ e @ƒY p A there is —sso ™i—ted —n exiom of g oi™e

for @ƒY p A stru™turesX —ny f—mily of nonEempty setsD e—™h of whi™h ™—rries

— xed @ƒY p AEstru™tureD h—s nonEempty pro du™tF st is n—tur—l to study the

logi™—l impli™—tions ˜ etween these —xiomsD —nd these m—y ˜ e quite su˜tle

even for the most ˜—si™ sorts of stru™turesD —s in ‘Q“F

yur p ositive results h—ve dire™t impli™—tions for su™h — studyD ˜ut not our

neg—tive resultsF por ex—mpleD let —n @ƒ Y p AEstru™ture on — nite set g

I I

P P

™onsist of sets e —nd f —nd — ˜ije™tion ˜ etween g —nd fij@e Y f AF vet —n

@ƒ Y p AEstru™ture on g ™onsist of sets e —nd f —nd — ˜ije™tion ˜ etween g

P

—nd fij@eY f AF „heorem W shows dire™tly th—t the exiom of g oi™e for

@ƒ Y p A stru™tures implies the exiom of g oi™e for @ƒ Y p A stru™turesF

I I P

yn the other h—ndD let —n @ƒ Y p AEstru™ture on — nite set g ™onsist of sets

e f

e —nd f —nd — ˜ije™tion ˜ etween g —nd fij@P Y P AF „hen we h—ve the

ro lem I oes the exiom of hoi™e for @ƒ Y p AEstru™tures imply the

exiom of hoi™e for @ƒ Y p AEstru™turesc

P

e neg—tive —nswer here would ˜ e stronger th—n the m—in result of ƒe™tion IQD

e f

whi™h s—ys only th—t — ™hoi™e fun™tion for the p—rti™ul—r f—mily fij@P Y P A

do es not gener—lly su ™e to dene — ™hoi™e fun™tion for the f—mily fij@e Y f AF

i i

nee™tive —rguments m—y pl—y — role in this new ™ontextF „o illustr—te how

this ™ould h—pp enD we t—ke le—ve of our ™om˜in—tori—l ™ontext —nd we ™onsider

—n ex—mple ™on™erning —n exiom of ghoi™e for — stru™ture typ e on sets th—t

oi™e for —re not ne™ess—rily niteF ‡e will show th—t the exiom of g

™om —™t —us or s —™es do es not imply the full exiom of g oi™e

„he —xiom e— „ ™ ono s—ys —ny pro du™t of ™omp—™t —usdor sp—™es

is ™omp—™t @where ™omp—™t me—ns —ny open ™over h—s — nite su˜™overAF st

is known th—t ‡e—k „y™hono do es not imply the full exiom of g oi™eF

‡e show th—t ‡e—k „y™hono implies the exiom of g oi™e for ™om —™t SH

—us or s —™esF sf ˆ Y i P is — f—mily of ™omp—™t r—usdor sp—™esD —nd

i

is — oneEp oint sp—™eD ‡e—k „y™hono —ssures us th—t ˆ a @ˆ A is

i

iPs

of D set ˆ a @ @ˆ AA  @ ˆ AF ™omp—™tF por —ny nite su˜set

i i

iPs iP

„he ˆ form — f—mily of ™losed sets with the nite interse™tion prop ertyF

ˆ a ˆ is nonEemptyF „hus

i

iPs

iven h—d we t—ken ‡e—k „y™hono in the ee™tive form th—t xesD on™e

—nd for —llD — ™hoi™e of nite su˜ ™over for every op en ™over of every pro du™t

of ™omp—™t r—usdor sp—™esD this —rgument still f—ils to dene — p oint in

ˆ F

i

iPs

‚ ‚ xg ƒ

e ™h—r—™teriz—tion of ™—n™el—˜le E—ry IF

stru™turesD €eriodF w—thF ungF @IWUSAD IUEIWF

’„heory of qroups of pinite yrderD4 hoverD xew orkD PF

IWSSF

ie™tive impli™—tions ˜ etween the ’nite4 ™hoi™e —xE QF

iomsD in ’g—m˜ridge ƒummer ƒ™ho ol in w—them—ti™—l vogi™D4 ve™ture

xotes in w—them—ti™s noF QQUD ƒpringerE erl—gD xew orkD IWUQF

RF e ‚ogersE‚—m—nuj—n ˜ije™tionD F

om˜F hF erF e QI @IWVIAD PVWEQQW F

ƒieveEequiv—len™e —nd expli™it ˜ije™tionsD F om˜F hF SF

erF e Q @IWVQAD WHEWQF

’f—si™ elge˜r— sD4 ‡F rF preem—nD ƒ—n pr—n™is™oD IWURF TF

UF ne theorie™om˜in—toire des series formellesD edvF in w—thF

P @IWVIAD IEVPF SI

VF gommuni™—tion sur les re™her™hes

de l— theorie des ensem˜lesD in ’elfred „—rskiX ™olle™ted p—p ersD4 firkh—userD

fostonD IWVTF

WF ’g—tegories for the ‡orking w—them—ti™i—nD4 ƒpringerE

erl—gD xew orkD IWUIF

IHF ’ƒhe—ves in qeometry —nd vogi™D e

pirst sntro du™tion to „opos „heoryD4 ƒpringerE erl—gD xew orkD IWWPF

’inumer—tive gom˜in—tori™s sD4 ‡—dsworthD wontereyD IIF

IWVTF

ƒieveEequiv—len™e in gener—lized p—rtition theoryD F om˜F IPF

hF erF e Q @IWVQAD VHEVWF SP