ro duing xew fijetions from yld @to pp er in edvnes in wthemtisA epril PPD IWWR hvid peldmn niversity of xew rmpshire tmes ropp wsshusetts snstitute of ehnology efeg por ny purely omintoril onstrution tht pro dues new nite sets from given onesD ijetion etween two given sets nturlly determines ijetion etween the orresp onding onstruted setsF e investigte the p ossiility of going in reverseX dening ijetion etween the originl o jets from given ijetion etween the onstruted o jetsD in eet neling4 the onstrutionF e present preise formultion of this question nd then give onrete riterion for determining whether suh nelltion is p ossileF sf the riterion is stisedD then nelltion pro edure existsD nd indeedD for mny of the onstrutions tht we hve studied p olynomilEtime nellE tion pro edure hs een foundY on the other hndD when the riterion is not stisedD no pro edureD however omplitedD n p erform the nelltionF he elerted involution priniple of qrsi nd wilne ts into our ruri one it is reognized s thinly disguised form of neling disjoint union with xed setD itself well{known tehnique tht ts into our theoretil frmeworkF e show tht the onstrution tht forms the grtesin pro dut of given set with some otherD xed set nnot in generl e neledD ut tht it n e neled in the se where the xed set rries distinguished elementF e lso show tht the onstrution tht tkes the mth grtesin p ower of given set n e neledD ut tht the onstrution tht forms the p ower set of given set nnotF uey wordsX fijetionD ijetive pro ofD qEsetD grtesin p owerD p ower set IF sntro dution emong the vrious mens of estlishing omintoril identitiesD mny omE intorilists esp eilly prize ijetive pro ofs4X diret demonstrtions y mens of mnifest ijetionsF he present sttus of the siene of ijetive pro of my e likened to tht of rulerEndEompss onstrutions prior to quss nd ntzelF iven with mny ingenious exmplesD ijetive pro of 4 remins n informl notionF he lk of suitle formliztion prevents exE plortion of the limittions of ijetive pro ofD4 nd so no results we know of my e interpreted to syD in ny senseD tht there exist nlytilly verile omintoril identities dmitting no ijetive pro ofF sndeed the sitution my well e sensitive to how ijetive pro of 4 is mde preiseD muh s nE gles my e triseted with mrked strightedgesD ut not with unmrked strightedgesF sn this pp erD we show tht just s lgeri metho ds @s emo died in qlois theoryA n e rought to er on questions of onstrutiility of geometriE l o jets y use of geometril to olsD so to o n metho ds from lger e pplied to questions of onstrutiility of omintoril o jets vi omiE ntoril to ols suh s ijetionsF e do not oer sp ei formliztion of the notion of ijetive pro of 4 in this pp erF ther we oerD nd exploreD distintion etween wek ijeE tive pro ofs nd strong ijetive pro ofs tht should e relevnt in virtully ny prtiulr forml ontextF vet us now explin this distintionF e omiE ntoril identity onerns two indexed fmilies of nite setsD sy fe g nd i iPs ff g D nd sserts tht for ll iD je j a jf jF por our purp oses nothing will i iPs i i e lost if we sp ek s though i were xedD so herefter we drop the susriptF xowD let @A e n ritrry onstrution tht pro dues new nite sets from old suh tht the rdinlity of determines the rdinlity of for ll F hen mnifest ijetion etween e nd f lso proves tht jej a jf jY we onsider this wek ijetive proof tht jej a jf jD insmuh s the ft tht the rdinlity of determines the rdinlity of is not itself proved ijetivelyF e strong ijetive proof tht jej a jf j must exhiit mnifest ijetion etween the sets e nd f themselvesF es we shll seeD for some onstrutions @A it is lwys p ossile to dene ijetion f X e 3 f just in terms of ny given ijetion p X e 3 f D nd I not tking into ount of the sp eil nture of the elements of the sets e nd f F rovided tht the denition of f from p is within one9s forml mensD one is then gurnteed tht wek ijetive proof using @A determines strong ijetive proofF por our purp osesD to sy tht the denition of f from p is within one9s forml mens4 is to sy tht f n e omputed from p D in senrio tht we will mke preise for the ske of deniteness @though the theorems tht we prove re not overly sensitive to the preise detils of the mo delAF hen the nture of @A p ermits us to ompute f from p D we sy tht suh @A n e eetively neledF e give exmples of onstrutions @A tht n e eetively neledD long with others tht nnotF wore generllyD supp ose tht @Y A is onstrution tht tkes two rguE ments nd tht g is some sort of strutured setD notion tht we shll mke preise lterF upp ose ijetion p X @eY g A 3 @f Y g A is givenF e derive riterion for deiding whether ijetion f X e 3 f n e dened solely in terms of p nd the given struture on g F his riterion is lgeri in ntureD nd in sp ei pplitions it devolves into questions out p ermuttion representtions of nite groupsF hen the riterion is stisedD nelltion priniple is gurnteed to existD nd we my pro eed to seek it outF sndeedD we hve typilly found tht when the nelility riterion is stisedD nelltion pro edure exists whose runningEtime is ounded y p olynomil funtion of the sizes of the sets in questionF sf the riterion is not stisedD then there is no p oint in serhing for nelE ltion pro edureF roweverD even suh negtive results hve p ositive imp ort of ertin kind for the working omintorilistF por instneD knowing tht the p ower set onstrution nnot generlly e neled suggests the e f p ossiility of lo oking for interesting ijetions etween P nd P even in situtions where no interesting ijetion etween e nd f is villeF sn ny seD questions out nelilityD when run through the mill of our generlEpurp ose mhineryD give rise to interesting questions out p ermutE tion representtions of nite groupsD nd thus n e resoure for lgerists in serh of novel questions to p onderF yur metho ds ome from group theory nd re purely mthemtilD s opE p osed to metmthemtilF xevertheless it my e fruitful to interpret our P results s sttements out omintoris in vrious top osesD though we do not pursue thisF yur work n e seen s following in trdition initited y edrino qrsi nd tephen wilne R in their study of the involution priniple nd ontinued y rer ert ilf IP nd fsil qordon S in their work on the priniple of inlusion nd exlusionF elso of some relevne is work of vszlo vovsz nd o ert eppleson on nite sets with reltionl strutures @see I nd the referenes ited thereinAF heir work diers from ours in tht we regrd two nite strutures s eing equivlent if they re denle from one notherD wheres those uthors require tht the strutures tully e isomorphiF xeverthelessD there re mny p oints of similrity etween their pp ers nd our pp erD in terms of oth metho ds used nd results otinedD nd it would e go o d to understnd etter the reltionship etween the two theoriesF yur pp er is orgnized s followsF sn etion P we give preliminries onE erning group tions nd x some nottionF xextD in etion Q we intro due proto ol to mke denite wht we men y eetive nelltionF hen we prove heorem ID the fsi iquivrine griterionD to formlize the intuE itive notion eetively nelle omintoril onstrution4 @etion RAD nd we give severl vritions thereof @etion SAF heorem P rests the fsi iquivrine griterion in more useful nd more elegnt form @eE tion TAF etion U digresses to tret some issues relted to vrint form of nelltionD for the ske of ompletenessF etion V shows tht issues of uniformity4 of nelltion pro eduresD whih re not ddressed elsewhere in the rtileD n e given preise formultion if one mo dies the frmeE work so tht the role plyed y tions of groups is tken over y tions of monoidsF he ve setions tht follow pply heorem P to otin onE rete resultsD some p ositive nd some negtiveD some new nd some oldX p ositive result for nelltion of disjoint union with xed set @oldD ut we oserve how it enompsses the elerted involution priniple of qrsi nd wilneD therey tting tht tehnique into our ruriAY negtive result for grtesin pro duts @oldD though not previously stted in this form in the litertureD t lest to our knowledgeAY p ositive result for grtesin pro dE uts with p ointed sets @newAY p ositive result for grtesin p owers @newAY nd generlly negtive result for p ower sets @newA whih is lried y the Q solution of n interesting groupEtheoretil prolemF etion IR gives slik pro of of very generl equivrine riterionF etion IS situtes our work in more strtD tegoryEtheoreti settingF pinllyD etion IT trets the relevne of our pp er to issues in the foundtions of mthemtisF yp en prolems re intersp ersed throughout the pp erF st is not neessry to understnd the foundtionl mteril in the rst hlf of the pp er to p eruse the vrious expliit exmples in etions W through IQY indeedD these setions ontin the min fruits of our workD nd the reder my wish to smple them efore studying the ro otsD trunkD nd rnhesF es middle ourseD the reder might strt with etions PD QD RD nd T efore pro eeding to etions W through IQF ome of our susidiry prop ositions hve trivil pro ofs tht my not e trivil for the reder to nd unidedY we hve inluded ompt versions of these pro ofsF roweverD rther thn try to follow these veritionsD the reder might nd it more helpful to drw the pproprite digrms nd hse rrows in the usul wyD onsulting our pro ofs only s lst resortF PF reliminries e denote the integers y D the nturl numers @inluding HA y xF ell other sets onsidered re ssumed niteF e qEset is set with n tion y group q @on the leftAF @por generl kground on group tions on nite setsD see ghpter I of TF sf r is sugroup of qD the osetEspe qar is lso qEsetD under the denition H H g