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38 K.A.JKesJiaKOB < -HaeaJi ( < -noflkojiluo). Bee BopaanoBH KOJiBua, paccwatpiibae - Mue B pafioie, npeanojiaraiotch He HwewmnMH ojieuehtob nopsmica 2 B CBO- HX aaantmbhux rpynuax, TO ecu, MS 2.x -O Bcersa ojieayet, WTO & 2 flokashmaetch, WTO JIOKOJIBHO paspeiuiimoctb H HOCTB MOpsaHosa KOJiBiia uobnaaaroi, TO CCTB ecjua aopjahobo KO.TIBUO JioicajiBHO paapeuihmo, TO OHO H JIOKB^BHO HMJiBnoienTHO. HanoMHHM, KOJIBUO (% HaSUBaeTCH JIOKaJIBHO HHJIBnoTeHTHtIM (,pa3peiflhmhm),ecjih JIB- <5oe iconeihoe MHosecTBO ero sjiewehtob noposaaet B Of HHJiHioieHTHoe ^paspeuimmoe; IIOBKOJIBUO. a sakjuo^ehhe 2 jokashsaetch, ITO BO BCHKOM KOpaaHOBOM KOJIBU6 Of CymeCTByeT JIOK3JIBHO HMJIBnOTeHTHhlM pa- ^, TO 6CTB JIOKajIBHO HHJIBnOTeHTHUM HfleaJI cx^,kotophji CO- Bce6e Bee JioKajiBHO HHJiBnoieHTHHe H^eaJiH 0Z, a $aktop- KOJIBIIO' Ot ~ & ne coaepsnt aaace JioKajiBHO paspeuiuwbix B 3 sonasaho, WTO ecjin MopaaHOBa ajirefipa ycjiobhk) MHHHMaJIBHOCTH fljifl HfleaJIOB, TO JIOKaJIBHafl paspeillhmoctb % BJieweT ee HMjiBnoieHTHocTB. OTCBfla cjieayet, WTO He cymectbyet npo- CTBIX JioKajiBHO paspeiiihmux MopztaHOBboc ajire6p, TO ectb ajiredp, pa^h- B cwbicjie BBeaeHHoro B 2 pa^hkajia <^f 4 nocbhmeh aokasaiejibctby HM^BnoTeHTHocTM Mopaanosa HHJIB - Ot (TO ectl ^JIH Ji»6oro BJieMenia <2<s Of. cymecibyei TaKoe WHCJIO n, K. (a), WTO CL^^O ), yaobjietbopaioinero ycjioshk) uakch- MBJIBHOCTH fljih no^kojieii. Sio yibepawehhe ododmaet Teopeuy Ajidepia C3] o HHJiBnoTeHTHOCTH KOHewHOMepHux MopaaaoBHX HHJtt-ajirefip. B [J2j A.M.lllHpmoBHM flokasana jiokajibhan HWJiBnoTenTHOCTB cneijhajibhhx MopaanoBux HHJiB-KOJieq orpahmwehhoro HHfleKca. BosHHKaeT BOnpoc, He dyayi JIM Taniie KOJiBEta H rjiosajibho HHJitnoTeHTHH? Kpoiae TOro, H B CBHSH c MSJioffieHHtiw B npeflhsymhx naparpa$ax soshmkaei ecte- CTBGHHHM BOIIpOC 06 SKBHBaJieHTHOCTH paspeihhmocth H HHJIBIIOTeHTHOCTH XOTH o'bi SJIH cneitiiajibhhx MopaaHOBHx KOJien;. Ha OTH sonpocu gaei OT- BeT noctpoehhtifl B 5 nphmep cnen;hajibhoro Mop^aHOBa KOJiBua Of : B Ot TOHsecTBeHHO BHnojiHeHH coothomehhh: G3=O H (a6)(ca!) = O, TO 6CTB O^M O, % «ectectbehho, JIOKaJIBHO HHJIBnOTSHTHO, HO Of H6 HBJIH6TCH HMJIBnOTCHTHUM KOJIBUOM. I. HeKOTopHe odntae CBOfiCTsa HanOMHHM, WTO HOpaaHOBbIM KOJIBIIOM (HJIH - -KOJIBUOM) KOJIBUO, B KOTOpOM TOHaeCTB6HHO BtfflOJIHHIOTCH COOTHOliieHHH: a 6-6a,
PaapeiUHMOCTB H HHJIBTIOTeHTHOCTB WOOflaHOBblX KOJieil, cjieayci, MTO ann.modbix sjiemehtob (2, 6, C & a? wop- KOJiBija To^eciseHHo BHIIOJIHHIOTCH (% - KOJIBUO, i^4"- ero afl/mthbhafl rpynna. BCHKHM 3Jie- '' nopoaflaet 3H.50Mop$ii3M fla rpynim (%*~ a.71? BCH- Koro 3C (X x &a = & >JHj;oMopa)M3a ffg HasbisaeTcn onepa- TopoM npasoro ywhokehhh (HJIH npocto onepaioposj ymhohemih) na 3Jie- MBHT O.. BceB03MOKHbie onepaiopu yuhomehum nopomsakt B KOJIBUG scex SHflOMoptpHSMOB rpynnu (%* noflko^buo /f, HasHBaeMoe KOJIBUOM npa- BHX y;,mo3icehh8 (MJIH npocto KOJIBD.OM ymhokehhm) Kojitua <^". Ma (I) M (.2) c^e^yet, ^JTO B KO^Bue ymhokehha fiopsahoba TOSSeCTB6HHO BhjnOJlHHBTCH COOTHOlfleHHfl: (3) BUTGKaeT H3 (I), ecjih B (I) ('i; BHTer.aei MS (I), ecjin B (I) HOJIOHHTB 6 = x., c=z. H ^=^» H (.5) GJiesyeT HS (.2), ecjin B (,2j nonosthtb ^ = ^/, c = Z H Q?= zc. IlycTB TenepB ^={-^oc}, oce^', - nekotopoe MHOTCCT- BO HesaBHCHMblX 3JieM6HTOB. MHOSeCTBO - " K3K-TO ynophflo^hm. PaCCWOTpHM KOJIBIJO KOMMyTaTMBHHX CJ10B OT 3JI6MeHTOB 31 K3K OT MHOKeciBa cbodoshux odpasybuuix. fljinaa caosa onpesejinetch odtiihhm CIIOCOGOM. ynophfloihm 3TH CJIOB3 T3K, 1TO CHOBa MGHBUieM fljihhu M6HBEie CJIOB do^biiiefi fljihiai; BHyTpa MHoscecTBa CJIOB oflhaakobom JJIHHH ynopn -, TO ectb ecjih Q. = TO a > cc.1 B TOM, CJIOBO ^2 HasoseM npabhjibhhm, ecjih a e X,, Kor^a, so-nepbhx <2,- H Q2 - npabhjibhbie CJID- Ba H, Bo-BTopux, C[f "& a z. MHOTCCTBO Bcex npabhjibhbix KOMMyia - THBHHX CJIOB OT MHOKCCTBa ^T 0(503Ha^HM_ ^ep63 X BosBweu MHomecTBO ^/={^^ \x- jc ] tpopmajibhux onepatopob ymh03s6hhh, HeflCTByKIUHX Ha KOMMyiaTHBHOM KOJIBIie_C MHOSeCTBOM CBO(50- odpasyioinhx X, H paccuotpim MHoaeciBo S/ -accouhainbhux CJIOB
40 OT ±7 ><a:c OT MHOKOCTsa fv.'»oo7;;;;;x o..gjim W = /fc^y ^^A, - Ky, TO ;;,. BSM AZ ^ ^--/ -,7j;;«ia cjiosa // * CC. /? - jyimho^ CJJOBO vv UHOseciBa ^ HHayuwpyeT ynophsohei-moctb MHO-, TO ecu, oy;iem cmstatb, TOO y. > ffy..ecjiii^/. ^^y- B X../nopHflOWHM Tenept MnosecTBO ^/ TJW, UTO cjiosa MeHBuieM / *- o'y^yi wehbiiie CJIUB uoramem /^ - MKHH, a BuyipH MHOKecTBa CJIDJ; coboii /P -/I,JIHHH ynophauyenhoctjb dy^ei.nekcincorpa&hqeckoa (B ynoph/jomehiiocth MHoacecTBa ^/ ). PaCCUOTpMM 3 CJIOB6 W H3 U nojicjiobo (F T) ecjin f? a < &c > TO if octanetch (5es 2) ecjiii /t7^? /P^, TO & samehhbtch na CJIOB: 3) ecjih Ka= c, TO CCTB a c, TO 2V sauehheich na cjie/tyiouiyio cymwy CJIOB: cootbeictbyioinyio no^ciahobky B W HJIH s2y/. Jlerxo HS no.tiyiehhhx GJI03 MeHtuie, wen W. li Kascaou HOM cjiobe eme pas paccmaipiibaem noscjiosa BMAS t/ H T.S. Bsjiay Toro, ^TO nojiyuaraaiiech cjiosa nueiot OTHOcmejiBHO Mnoscec- 3Z Ty me /yihuy a TOT ace cociab, mo H CJIOBO NV, a CJIOB, MeHBuiax, ^ew w, JiHuiB KOHeiHoe IHCJIO, TO na icanom-to mare, UTO C.TOBO NV (HJIH SJIH noflxoafimero qacjia x< 2.K\*J ) ymaok CJIOB K6T H OTCyTCTBOBaTB. CJIOBO ^/ BUfla (6) Ha30B6M J2" -HOpM3JItHbIM CJIOBOU. riycib Tenept C% - KOJIBUO, o& - ero OO^KOJIBUO, fl - KOJIBUO
PaSpeffiHMOCTB H HHJIBnOTeHTHOCTB BopsaHOBHX KOJien. KOJiBita Ct. Toraa ofiosnaihm Me pea H tjepes B - noskojibuo KojiBqa /?, nopojcaeiraoe &. B GHJiy (4) H HsJioseHHoro BHiae ouebhano, 110 cnpaseotmba wie- JIEMMA I. II y c T t ^ -^-KO^Biio,^ -ero n o a K O B B u o, JT - U H O s e c i B O oopabyromax dzr.torfla jirocoa ajiemeniw HB^ HJIH,; JJIH noflxofla mero lac^aa', ^^w, p a B e H J I H H e f t H O t t K O M f i H H a U H H ^ T - H O p M a J I B H U X CJIOB T O P O ae coctasa O T H O C H T e j r t H o J r. flokaacew TenepB npejuioiehwe I. IIycT!,<iz5-n/C -HaeaJibi^T o a e I) HycTB ^",, Z ^ <55- ; a e (%. Toraa no (I), ya, za e 5. 2) HyciB & XrjCt&e/C; ae (X. Toraa no (I) Ho [csa)d~\c, Cca)(Saf)t (da) (6c)e (& ), a \ ca)d~]6, (fa) (cd) e & ( ) ; Ho \S(ca) ~]d, [6 (daj]c, (6c)(cfa),
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PaspefflHMOCTB H HHJI&nOTeHTHOCTB JiOpflaHOBHX KOJien, H KOJIBIJO (% - HHJItnOTeHTHO. CjieflOBaTCJIBHO^H KOJILtlO Of MJItnOTeHTHO, T3K K3K 0?^- &? Teopewa aokasaaa. CJIEflCTBHE. II y c T B <#" -j7-kojibu.o,<*r- ero nojj.kojibu.0 H o# - J I O K S J I B H O p a s p e m H u o. T 0 r fl a o# - J I O K 3 J I B H O H H J I B n O T e H T H O. JIEMMA 3. ECJIH K O J I B U O & H M e e T K o H e i - Hoe gucjio odpasyjomhx. To a Ji H Ji to <5 o - ro imcjia/tz. ^/rz Tose Hweex KOHeinoe IHCJIO ofip a^ y K) m H x. HoKaseM, ^TO C%j HMeei KOHe^aoe IMCJIO odgasyiomhx. IlyciB 5T - MHoaecTBO o<5pa3yiomnx KojiBua CZ, WHCJIO B 5T - /Z. Ho flokasahhowyb jiewne 2 cymect^yei TaKoe ^HCJIO f(n,2), ^Z ' <^2. /lokaacem, ITO IHCJIO ocpasyromhx KojiBua me ' x pasjihirhx KOMMyiaTWBHHX CJIOB fljihhil /C ^J)T /Z_ CBOofipasyioiiuix. B cauoii aejie, ecjin aaewent ^ HS & Hweei fljinay OTHOCHTeJiBHO MHOscecTsa afega t- j-(n,2), TO, no JieMMe 2, r /^f **~ s^* {/ ' s) CZ f, TO ecib pasea cymme ajiewehtos, Kas^a HS KOTOeciB npohsbe^ehhe ne wenee ICM Tpex sjiewehtob H3 (% B KaiecTBe ospasyronhx KOJiLua %/ MOSHO BSHTB HeKoiopoe CJIOB OT X, JIHHa KOTOpUX OTHOCHTe^BHO JT MeHBffle, ^614 f(rt,2) Ho T3KHX CJIOB MeHBIHe, VGU M. flokasaiejibctbo nemu BasepinaeTCfl oqebhflhou HHayKuneM no JIEMiJA 4-. ECJIH y - J i O K a J i B H O paspenih J I O K 3 J I B H O p a 3 p e i U H M O, T O HKOJIB- / x/ U, 0 {si J I O K 3 J I B H O p a 3 p 6 I H H M O «IS & y HBJIfleTCH JIOK3JIBHO paspefflhmum HfleaJIOM M Tosce 6yaeT JioKajiBHO paspenihmum. flokaacea, ^TO Ct K8JIBHO paspeiflhmo. B03BM6M B KOJIBU6 C% noflkojiiuo ^, KOHGIHUM»JHCJIOM SJief-lBHTOB n.. B $aktop-kojibh6 ffi Odp83 ua_xt - noflkojibrio HMeei KOHeiaoe IHCJJIO odpasybmax ^H pasperauo. IlycTi ^(p) O Toraa B KOJitoe (% (p) ^ Ho, T3K KaK?B npekhhx 0003HaienHHX7/^) S (n), TO >O) >". no Jieuwe 3/C^» raeet Kone^Hoe IHCJIO odpasyiomhx^ H noatouy no Teopeue I no KpaftHeH Mepe "({/-n /^.^yr&»to ecib KOJIB-
UO - paspeilimmo. TeM CaMtW?BBH y npomsbojibhocm KOJIBUa /C", JO- Kasana JiOKaJiBHan paspeujwuoctb KOJitita C%. 3Ha^HT,JiOKajiBHO paspefflhmo H KOJIBU.O (%, T3K K3K Of ^&. npejuioiehme 3. C y M M a flsyx J i o K a j i B H O p a 3- u i H M H X H f l e a j i O B y - K O J i B a a (%. fiyaei ajibho p a a p e n i H M H M Mfleajiou. B camom aejie, nycib tyf K U% - JioKaJibHO paspeuihmhe H^eajiH. y-=yi+yz. Toraa 5V /? ^ H Jz/y^n^ JiojcaJiBHO paspe -. Ho %r, a= ~^2/y nyz- OiCBaa H HS aeumh 4 cseayet JioKajiBHaH paspeuihmoctb H^eajia ^7. OqeBHOTOii HHayKUHeK OTCKWa BaBOaHTCH CJIEHCTBME.C ymma Ji»<5oro KoaetiHoro IHcjia- J I O K 3 J I B H O paspeiflhmtix HfleajioB psjifletch jiokajibho p a s p e r a n u H M use- JI 0 M. IIPEflJIOSEHME 'f. C y M M a scex JionaJiBHO pasp e m M M H x HjeaJioB^ -KOJiBiia <^ dy^ei j i O K a a B H o p a s p e u i H M U M H a e a n o M. IlyciB o - cywma scex JioKajiBHO paspeiiihmhx ajeajiob KOJiBita <^ in.ai,aij...)ctrl - KaKne-To ft sjieuehtob as c. Toraa JJIH siofioro i.jc**-/,2, >,K- > cymectbyiot Tarae JIOKBJIBHO paspeuihmtie see MHoacecTBO ajiemehtob^v^qa,...^^^ nphhaflfleatht cywue qwcjia JioKajiBHO paspeuihmux HaeajioB, Koiopan cawa HBJiHeTCH paspeaihmtim H^eajiou. CjieaosaTeJiBHO, noakojibito^ nopowaehhoe w&at,q2)...,&ft. paspeuihmo. Ten camhm aokasano, ^ITO. - OnPEflEJIEHHE. JIoKa^BHO HHjiBnoienTHHft HfleaJi a ^ -KOJiBua <# Hasosew JioKajiBHO H H J i B n o T e n T H H M pajh - K a Ji o M, ecjih OH cosepsnt see JioKajiBHO HHJiBnoTeHTHHe naeajiu K03IBU3 C% HftaKTOp-KOJIBUiO Oi = / H6 COflepKHT CoCCTBeHHHX JIO- K3JIBHO HHJIBnOTeHTHHX mo cywiia ol Bcex JioKa^BHO paapeidhmux OKaJIBHO HHJIBnOT6HTHHM pashkajiom. B C3MOM S6JI6, oc - JioKajiBHO HHJiBnoTeHTHHK Hflea^ no npeanohehhio 4 H Teopeue I M <-% He coflepmwt codctbehhhx JioKajiBHO paspenihmhx Hfleajios no neuwe 4. MTaK, Hueei MGCTO TEOPEMA 2. B Ji»(5oM J -KOJiBue^T cymect- ByeT J H O K a J I B H O K H J I B n O T e H T H H S p a a H - K a Ji <
H HMJiBnoieHTHOcTB iiopaahobiix KOJieii 47 3. JIOKaJitaaH paspeimraoctb H HMJitnoTeHTHocTB yjobjietbophkinero ycjiobhio MUHMMMBHOCTH SJIH JIEMMA 5. II y c T B B ^ - K o ji & q e " n o fl K o Ji B- U o d r jioicajibho p a a p e i H H M O. T o r s a B K O J I B U 6 y M H O K e H K g / 7 7 noflkojibuo^/5 JIO- K a ft B H 0 H H J I B n O T e H T H O. IlyciB B czr- noskojibuo /C" noposcaaetch Kone^HbiM MHoseciBOM 3Jie- M6HTOB., nyctb B X n, 3JieM6HTOB. II 0 TCOpeMe I KOJIBUO/C" HHJCbnoTeHTHo; nyctb HHaenc ero HHJIBnoTeHTHOCTH paseh m. : /C'^z=6). BsHfly 1.4), B KOJiBue /7 HOOTOJIBUO 7, noposcaehhoe MHoacecT - BOM /? = \/i?c /7/ce }, noposcaaetch KOHeqHMM MHcmeciBOM TOB y>y~{3.xlxe B03BM6M Tenep MS C, rje a ^ - MMCJIO pasflhihhx KOMMyTaiHBHBIX CJIOB fl^hhh A OT AZ. ospasykiimx. IIo neuwe I w pasen cywae ^ZT -nopmajibhtix CBOB BHx H oicytctbobaibj. IIoaTOMy Jiiado afet/k * m,, JIH- 60 cleg ' ZK t- m.. Ho Torsa B KOJIBUS JiH(3o. -?, O. CjieaoBaiejiBHo, jih<5o /?y= O, jin<3o /%^. = o H 7 H C^= O. Tenept MXSoe KOHe^Hoe MHoiKecTBo Z ajiemeniob us Torja MoatHO yicasatb Tanoe KOHe^noe MHOKeciBO onepatopob MS %.., MTO nopoikaehhoe HMH B /7 noflkojibuo coflepacht Z ; no TOJIBKO ITO aokasaahomy 310 KOJIBIIO ^OJIKHO CHTB HHUBnoTeHTHHM. 3na-, KOJtBUO B JIOKaHBHO HHJIinOTeHTHO. TEOPEMA 3. ECJIH Of -JioKajiBHO pasperan - ^? -ajiredpa naa noaem^",yaosjie - pfljoiah ycjiobhk M H H H M a J I B H O C T H aaeajiob, TO ^ HHJiinoTeHTHa.*-' TeopeMH octaeicfl cnpabesjihsuu H B TOM C/C HBJIH6TCH <V -KOJIBUOM. a
48 K.A.iesjiaKOB (% we pea CP&i, "-wepes j-o H, ecjin yjfce onpeaejieho, ((Z^OljOt + (%^(%(%.) - wepea Ife npefljioxehhh I cjieayet, WTO RJIR jnotforo wucjia /c JIH6TCH HfleMOM B (25T. yfiubaioiuyio uenowicy.-. (ID B cuay ycjiobmh MHHHMajitHOCTM aa KaKOM-io mare rt 3Ta iienoqka cia {5HBii3HpyeTCH, TO ecu, (Tpj=<^+yf " =<3[?z+ f ' ' ' ECJIH <^'[:/zd=^7 ' TO see ysce aokaaano ^BBHay npejyioffiehmh 2)., WTO B chjiy ycjiobhh MHHHwajiBHocTH, cyntectbyei TaKoK, < ^O, ^^^j-^-j, WTO U2) (&(%)(% +&QXGL) =&. (13) cjie^yei, WTO SJIH Jiiodoro ajiewenta «5 MS os' cymectbyiot ia- H T3KH6 3^eMeHTH 3CK,UK Z K e. (X, WTO ecjih s WHT, wepes wepes, nopoaaehhuk B OZ. ajiewehtamh cf)c2,---,cp > MOSHO pac- K3K /7 - MOSyJIt ^^ C OdpasyiOElHMH C<,C2,...., Cp., nopohaehhhk B ^ ajieiiehiamh Cj,c'2,...,Cp, UOKHO pac- KaK /7 -MOflyjiB^f/ c odpasyiomhmh Cf,c,...,Cp. HCHO, WTO ffft'^fftt./*?. TaK K8K /7 - pashkajiihoe B ciuiy 5 (aase JioKajiLHO HHJiwioTeHTHoe; KOJIBUO, a Wf. - KOHGWHOIIO - /7 -MoayjiB, TO, no H3B6CTHOM Teopewe (DUi TJI. yill, 4) c: ^?X. IloaTOMy fflk'^fitz. A T3K icaic ^'e 55-, TO, saa- TenepB ^"'wepes H, ecnh^r^-j yace l^3 Toro, WTO <& co CBoHcTBawM (.13),, H MS MH-
PaSpeiUHHOCTB H HHJItnOTeHTHOCTB KopaaHuBHX KOJien. 49 B chjiy ycjiobhh MiiHHMajibHocTH 3Ta ijeuoika aa KaKOii-TO A -on neexe floji»aa odopsatbch:. &? ', =O.> SHaiHT, cyiuecibyet TaKoe IHCJIO m, rrz^2/<, ITO win JIBO'Oro ajienehia a MS ^2'/ a Jiiodoro CJIOBB W HS /7 / -fljmabi ne MeHiuieft ien /rt., Hueei uecto: a\*j O Tan KaK us TO H Z^W ~ O. H as (4) cjieayei, ^TO BJIH jnofioro ajieuehia Se-ofar cy- IHCJIO m = iaji ^^, TO a. npiiha&nemaji OH H <^s SJIH jiio(soro ihcjia 5. A Toraa miejio <3H aecio:<i5'^r = ^7 (BBH^y (15)), npothbopeiht (13). SHa^HT, MHOK6CTBO 5C aenycto. _ PaccuoTpmi $aktop-aare(5py (% = CK2. TaK KaK (% TBOpHei yc^obhb MHHHMaJiBHQCTK win HfleaflOB H HueeT Hy^esoe ymhojte- HHe, TO Of. - KOHe^HOMepHan aarefipa. nycti ^? ^Zl...,eK } - ee 6a3nc._ BOSBMBM no npoodpasy Kaaaoro Q *(% & H odosaauhm MHO- KecTBOj^,^,...,^} lepes j 0- Q noposaaei B Ot Hyio, B CHJiy Teopeuti I, noaajiretfpy. FlyciB HHfleKC HOCTH 3Tott noaajiredpbi pases t : J&=O. PaccMOTpiuj (%t. 2oKa»eM, ITO ecan ae.0f*\o B caiaou aejie, G=^^(X t fae Kas^ng a<; BCTB Hue comhokht6jie8: ^. TO CnOCO(30M). B03MOKHH 2B3 CJiyiSfl: JM<50 KaKOK-TO Ot& H Toraa aaeweht a^ nphhaajiesht (%*-*', ^adp B 02HOM cweshom K^acce no ^20 KaKHM-TO 3'neMeHTOM ^j,. us TO ecti j:.. =A.. +/-, rae /. e-cx2-. B STOM </c v V v ectb npohsseaehhe t s^emehtob, no KpaHneM Mepe.oflHH HS KOTOPHX nphhafljiejtht O z. Ho Toraa a^ecz** HH^BnoTeHTHOCTH aotecjpii. MiaK, see, M sneiieht a. nphhaanesnt CZ^*'. BBH cjieayet, MTO (% ^ = CK^'. EosTowy Zf
50 K.A.lGBJiaKOB aokaaceu, UTO HUH mosoro ajiewehta a pajibhoro ihcaa /? tmeet MCCTO 5 us H aiofioro naiy- (17) HOK3K6M (17) JiK)6oro < HO /?. = / 3TO PaccMOipHM aro-,. ap-,~ap^,-i., r«e HH6 /?-/ 3JieM6HTOB H3 ^ : dp-t}i pacctasjiehh KaKHM-io cnocodou). BOSMOBCHH TO JCve, (%z H Toraa ^o/, ^ ^ ^^, JiH6o OflHOM CM6SCHOM KJiaCCe no ^2C K3KHM-TO /?-/ ajienehtob, no KpafiHeM <^2 ; HO Toraa a/,&c%p IlyCTB e ap~*. (18) GCTB : Jiwdo Kaicofix.- JieEHt B s-. 3 OBHH MS Koiopux M3 (18), 1TO K (17) flokasano. B wacthocra, CTaBMTB B BCSHCMW anewem MOKHO npea - ; - TO se, UTO B (16)). (19), TenepB B ajiredpe ywhosehha Jl noaaaredpy <, no- MHOXeCTBOM Rg=\ftx ji\x<s <S. TaK KaK ^ - KOHei- HonoposcaeHHaH HHJiBnoTeHTHan ajirefipa, TO $ - KoneiHOMepHa, noa- TOMy ajiredpa 5 noposaaetch KOHeiHHM MHosecxBOM odpasyroipdc H, 3H3WT, no newme 5 5 HHJitnoTeHTHa. Ilycii. HHjeKC HMJiBnoieHTHOCTH < paseh jy, TO ecti> 5 = ^7 (20) Teneps B JJ CJIOBO W R. - SJIHHH H KaacaaM J^, /<=/72,.../yv', npez(ctabhm B BH^e (19), ITO W^. ECJIH KaatflHft -2",v ^^» T0 ^i~ no '[-TO 3JI6M6HT Zjc Ct*\, BBHHy (16), ZfL ~ ^t juofioro HaTypaaBHoro TOcaa S. Ho Toraa as (15) H TOPO, ITO oo" - Hjeaji KOJiBua (X, cneayei, HTO uw^ o OTH jnodoro ~ "^
II H.-iJIHTOTeHTHOCTB ilooaahobux KOJI6IJ 51 TO ecrfc cjwt<=<?. Tea ITO {(.< &)& ' }<% =0 v ' iio 3 TO npothbopeqiit (1.3). SuaiHT, npeanojiosehhe (12) BG^CT K npo- THBOpe^HK, M H3 C3MOM 36Jie C%ty^ O Ho Toraa no npefljioacehmo 2, ajirefipa <^ MJiBnoTeHTHa. Teopeiaa flokasaaa. GJIEZICTBME. He cymecibyer npociux Jio- K a J i t H O p a s p e n i H M H X g o p a a H O B b i x an - r e 6 p. *) HHJO-KOJii>ua,yflOBJieTBOpflK)mero MaKCHisaJiLHOCTW TEOPEMA 4. HycTLiT -^-HHJII. - K O J I B U O, y a o B x e T B O p n i o m e e ycjiobhb M a K C H M 3 J I B H O C T K ^JIH HOflKOJieil. TOT^a ^F H M J I B H O T e H T H O. T3K K8K MHOSeCTBO HMJIBHOTeHTHBIX noakoji6e[ HHJIB-KOJIBUa CZ CO no.hkojibao, noposcaehnoe ajiemehiou &G.CP, TO B (% cyme MaKCKwajiiHoe KHJiBnoieHTHoe noflkojifcuo *2/Jf, ^ ^ O. npothbhoe: Toraa cymectsyei TaKoH ajieueht a0.o, ITO, JiH(5o Ha ajiorhihhe paccy»sehhh HOBTOPHBTCH H T.fl. TaK KBK ^ yaobjietbopnei ycjiosiro MaKCHMaaBHOCTH jyia., TO Kasfloe noflkojibuo (^ HJieeT KoaeiHOe IHCJIO odpaaybinhx, B, H 9ft. Toraa B KOJIBUB ymhoxeaaii J) nojkojiiuo //,no- MHosecTBOii % r[/?x -/?\z>e'ffl' j, no t^) TOKC HBJIH- KOHeiHonopoaweHHtiM H, no JieMsie 5, KOJILUO HHjeKC ero HHJiBnoieHTHOcTH pasen Jf. Ho Toraa a^ -Q0R.Xff^Xz... ^fj^oe<^. nosiouy TSJIBHOCTB 3JieM6HTOB ao,af,cz2,... KOHeiHa H cymecibyei lakojs 8Jie- M6HT O., 1TO a^"^» HO CtXeVK 3JIH 3IK)6orO 3neM6HTa ^T ^ 2f. PaccMOTpnu sjieweht ^2:^jc, aze. 9ff. BOSMOSHH asa ciyian. yrsepsflehhe cnpaseajihbo soodne wm
52 K.A.IesjiaKOB Bcex Toraa no (I) H a3xe9fc SJIH scex xcvtf. B CBMOM H xa, z*z;<2r e ^f, nonojiywaetch, TJTQ jno«5oro ^ e W. HJIH JiioGoro ihctia A p p(a) TaK KBK ^ - HHJIB-KO^BUO, TO cymectsyet TaKoe, ITO ^^ 7, H nostouy cymecibyei TaKoe IHCJIO «TO a^^$jf, a (a*)*ew. Tor^a ^Aa^a^-2, HO KO ^TO <5uao aokasaho.qio STO ajiewent HsJ%.BooCme ( naiypajilhoro ihcsa /c>/ a: e ^. H c&cew MR raofioro OtfosHaiuM STOT s^eueht a lepes d 2) CyiecTByei Taicoa sjieweht xefyz, ITO a x BosBMeM npohsbo^bhhtt BTOueHT ye% H paccmoiphm Ho (2) ITO e ajis jno<jhx x,yew. (21) PaccMOTpmi TenepB (a&x)2'. no coothomewib (2) \^(a x)x~\a.}a, n ^ax)a~\(a2z) By 3jreueHTa <2:. IIO COOTHOffleHHK) (I): nphhaajieaat 2^ no 121) H TBK y.'k&\ascxa]\x e. Vtf no (.21). no (21). c^eayei, quo^zjc^f.noaiomy no HflyKaHeK no K (H^H B cajiy (4-)) HaxoflHM, wio ro HaiypajiiHoro iwcjia x>/ ^a^^e1^. OCoanaiiiM STOT 3JieMeHT azx yepes S., B odoiix cjiywanx Ha8aeH TaKoH sneweht HO 507 e ^f,zwih jimfioro jr " 2^ H HJIH jno<5oro HaiypaaBHoro MH-
PaapeiiiHMocTB H HMBnoTeHTHocTB fiopaahobhx Koneii 53 K> / S. oflkojibno, noposcaaewoe B CZ *%? H 3 lepes., HO ^f^ e^f. T3K K8K $f HHJIBIIOTeHTHO, TO OTCKJfla, mo KOJIBUO 2^ paspenihmo. Ho Taic KaK KOJIBIJO ^. raeet KO- HeiHoe tihcjio odpasyioiitfix, TO j^, no Teopewe I, H HO, 3TO npothbope^iht MaKCHMajIBHOCTW JIOflKOJIBIja Ciaso (5HTB, tyfl=(%. TaKHM ospasom, KOJIBUO MTO H TpedOBaJIOCB 5. npraiep cneiwajibhoro MopaaHosa paapeinhworo, HO H6 HHJIBnOT6HTHOrO *ffl UieHHH paspeiuhmocth H HHJIBO OT6HTH OCTH 'Y -KOJI8U B OfiieM CJiywae HJIH B CBH3H C H3JIOTCHHHM BUIfle HHTepeCHO BHHCHHTB BOHpOC 0 COOTHO- B cayiae cneuwanbhhx ^ -KOJieit. KpoMe Toro, B ^2] jokasaho, UTO BCHKoe cneuwajibhoe 'J -HMJH>- KOJIBIIO orpaahqehhoro HHseKca JioKajiBao HHjiBnoTeHTHO. UnTepecHO BHH- CHHTB, ne dyaet JIM TaKoe KOJIBUO H rjiodajibho HMJiBnoieHTHHii. Ha 3TH Bonpocn saei OTBBT nocipoehhua nasce npnuep cneawajibhoro MopaaHOBa HH^B-KOJiBiia HHfleKca 3, KOTopoe paspeuihmo HHfleKca 2, HO He HBJIH6TCH HHJIBHOTeHTHUM. HanoMHHM onpeaejiehhe cneuwajibhoro MopflaHosa KOJiBqa. IlycTi, (U. - accouhathbhoe KOJIBIJO. Ha ero aashthbhoft rpynne onpeaejimi HOBJOO onepamim o 'J -ymhoscehha: cto$ as + 6o:. OT- HOCHT6JIBHO 3TOPO HOBOrO ymhotchhfl H CTapOPO CJIOK6HHH HOJiyiaeTCH KOJIBUO tu. ^. -KOJIBUO (% HasusaeTCH cneiihajibhum, BCJIH cymectsyet Tanoe KOJIBUO <$, UTO (X H30MOp$no nekotopowy noakojibdjy (tt B EpOMSBOJIBHOM KOJIBU6 7L HKO(5HaHOM 3JieM6HTOB (2, 6 H C H3- upascehhe / (&} $? c) (_a6)c +(6c)a + (ca)6. CTpOHTCH nphmep KOMMyTaTHBHOrO KOJIBIia, B KOTOpOM BC6 HKO- 6H3HH TOSaeCTB6HHO pabhbl HyJIK. nyctb X=\Xi,Xz,...,Xn,...j - cieraoe MHOTCCTBO HesaBHCHMtK CHMBOJIOB. MHoacecTBO X ynophflotihw: HOJIOKHM ^ ^cr,-,ecjih c</. Jliofiyio Henyciyio nocnesobateabhocti sjiewehtos M3JT aasobea cao- BOM. fljihhok CJIOB3 U. HaSOBBM UHCJIO Ct(u) BXOSCfleHHg B U 3JieM6H- TOB MHosecTsa X. CJIOBO u nasobea npashjibhau, ecjia jindo TO ecti d(u)=-f, JMdo B TOM c^y^ae, Koraa ^ xc^^c cflosa ynophflohhm j[ekchkorpa$hweckh, TO ecib nojio - ra-
K.A.KesJiaKOB npmcoeflmhhm K MHOKeCTBy npabhjibhblx CJIOB CHMBOJI O H noct3bmm B COOTBeTCTBHe JHOdhlM flbyu UpaBHJIBHUM CJIOB3M U KOMdMHaijHio u * V 1) LL* O O *u =O ; 2) L npashjithbix CJIOB no cjieayrainhm H V X I / / ^f. **f* ^y** ^// A^*1TH// *"* ^**'1 ~V W 7TTTC7 H^tfO «-^ y <A ^^ ^C>t- ^~" -*- >fc* " OC < COJ1W // ^^ ^/ * ^L * - x ** ih"** tlcftw 'N /\ *- * "^O ^ ^.p * * * c- /^ H" J c^c Jf 3 ^~ ^'tf " C*C ~ ~ C^ i 6 CJIK QL\ *./ ^ / n * ^ *^ /c f 5.) 3 6) - npabhjithoe CJIOBO OT ajiewehtob a <5 - qhcjio HHBepcMg B nepeciaaobke (<lf,c2,...} Ln^/<). ECJIH aaaa napa npasmibhux CJIOB u m w, TO, nepexoan noc - jieflosatejilho OT npa^hji c uehtuihuh HowepauH K npabhjiau c doji&imuih HowepawH, rje-io BCTPBTHM npashjio, nokastibaiomee, KaK JJIH aahhux u H V odpasobatl u * v. Bee npashjibhae cjiosa (c O ) OTHOCHTeJn.no ymhoscehhh * H $opcjiosehhh odpasyiot KOMwyiaTHBHoe KOJIBIIO Of, B KOTOPOM see T03KfleCTB6HHO pabhh HyJIK). flesctbhtejibho, yuhosehhe K KOMMyiaTHBHO. HystHO eme nposephtb, 1TO /(U,V,\S)=O OTH BC6X U,tT,\^ K3 &. floctatoiho B cnjiy jihheiihocth HKOdaaHa upobephtb BTO SJIH cjiyiiah, Koraa u, V> w - npashjitatie cjiosa. ECJIH XOTH da OSHO HS CJIOB u, T t w pasho 6> HJIH fljihha XOTH 0H 3ByX H3 CJIOB U, i/, W SOJIBlUe I. HJIH B CJIOsax <^, t^, W ectb odnyie ajiemehtu HS 3T, TO j'(u,tj,\j) =-O TpHBHajitHO, T8K KaK Kas^oe cjiaraeuoe pasho O. To dl(u) > /, & =.; } W-ZK H JIHfiO U < Xj, JIHdO U < XK. pasodpaib ssa 2) U-Xcj <S=Xj, W=X* ; Z >jcj>3c WT3K, I) UtCZ.-KZ/J^UtCZ.'XxJ^ O» TaK K3K
H HHJIBHOTeHTHOCTI) MOpflaHOBHX KOJI6U 55 ftllft { Cf(.U), Cffy H (u* X;) x XK=-(u* oc*)* X; /iehctbhtejibho, nyctt V V IHCJIO HH3epcnM B nepectahobke 6v,?,, n,/) pasno p H <*& xtz -"^cn */ > - npashjibhoe CJIOBO OT se 9 X2J..., x^n, Tj,, TO ecu. nocjieaobatejh>hocti> BTHX ajieuehtob, aanncahhafl B nopfqpce BoapacTaaHH MX HHfleKcoB; 3Ty MOHOTOHHO B03pacTa»myfo HOCTL HHaeKcoB o(5o3haihu lepes < i*,, cs,..., ^,4' > IHCJIO HHBepcafl B nepectahobke (<if, a,' ',,1 />,x) pasho y.toraa (a * Xj) *X*= (-f) x... x : K> - npabujiihoe CJIOBO OT -r*. UHCJIO HHsepcHg B nepeciahobke (V,, ^,...,4t,* cjiy MHsepcHii B nepeciahobke (<,,..., in,/?, A ), Tan Kan K>/, H(5o no ycjiobhio ar ' > 3CK, H A HHBepCH8 c J He ofipaayei. TIo- HHTHO, ^TO ^HCBO HHBepcua B nepectahobke C< i,j,...f c^, K>, / ) pabho /?.+ /, iak K8K / odpasyet c /c oflhy HHsepcHio. = o * Z = 0. HT3K, (^-flekctbhtejibho KOMMyiaTHBHOe KOJIBUO, B KOTOpOM TosflecTseHHO pashh Hymo. ByaeM Tenepi odosnaqati nponsbeajiewehtob a H 6 HS (% npocto tepes 06. HCHO, MTO JiioOux 3JieweHTOB a, &, c, <3^ us (% ToatflecTseHHO BunojiHrnofCH : (as)(caf) O M a3 O. C apyroa ciopoau, (% He HHJiBnoTeHTHoe KOJII.UO: npohsseflehwe Jiiodoro BC6
56 K.A.SesjiaKOB 3JieM6HTOB H3 3TC- }X; _,...,-2^-..., B3HTHX B nopfl^ke HX HHH, 6CTL npabhjibhoe CJIOBO 3?i XL ^f- ^ O. Ilpesme vein aokashbatt, MTO C - cneimanbhoe -KOJIBIJO, ao- Kamea cjieayiomee IIPEflJIOIEHHE 5. H y c T B - a c c o u H a T H B H o e K o Ji B u o. O n p e a e J i H M na a ' a a H T H B H o H rpynnekojitaa/^ nosyh) onepaijhio x : a. x = av 6<z\e KOJIBIIO tor - H O C H T e J I B H O Ciaporo C J I O K e H H H + H H O - s o r o y s j H O s e H H H x ) o ( 5 o 3 H a q H M q e p e+s/7 ~. Toraa scakoe n o a K O ^ B a o KOJiBHa // 'n B- JIH6TCH C E e a a a J I B H H M ^-KOJIBqOM. Ha aaanthbhok rpynne KOJiBua /7 onpeaejihm onepai^hjo & - = -a-&. IlojiyweHHo HO, (a--^)-c =(~&&) -c = a&c flcho, 1TO KOJIBUO /7 HSOMOpfHO KOJIBlty >7, H HOTOMy K8KnoaKOJiBijo Ko^tija >7 'HSOMOP$HO cootbetctbyromeuy noflkojibqy KOJIB- Tenept KO^BUO (X H KOJIBUO >7 ero ymhosehhli.b HDJIB- MHOTCCTBO ^* Bcex onepaiopob HBJ1H6TCH rioflkojibqom. B C3MOM T3K K3K OTH JIK)(3orO TaK ^ITO OTHocHTeflBHO onepauhk -f- H x - MHOKeciso scex onepaiopob npasoro ytihosehhh - HBJIHSTCH KOJIB^M., ITO KOJiiua & H ^ H30MOp$HH.. IIOJIOKHM <^(aj=^a. no TOJIBKO <JTO aokaromo«op$h3mom. Jaaee, ^f(o) J-}0-O, u 06- paiho, ecjih (f(a.)-o, TO H OC-O flesctbhtejibho,nycii Qf^ O H nycib OTH sanach Of. B d5f HyKHH ajiemehtu^^,...^-^ H3 3?, TO ectt <or HBjmeTCH HeHynesoa MHeKaoK KOMdHHauHeM HCKO- CJIOB OT ^;- }Q,- & Bo3BM6M Tenepi. TaKoM 3Jie- M6HT XK M3 X, 1TO OCK<3Ci,,, j = 1,&,.. -, 7Z. Ho T0r«a - H 3TO onflib HeHynesafl KOMdHnauHH (c TeMH see K09$$HUHeHTaMH) npabhhbhux CJIOB. SHaiHT, d? O BB6ieT 33 CO(Jofi ftcz * O. TaKHM odpa30m;hapo V Ke? V> = O
PaspeuiHMOCTB H HMJiBnoieHTHOCTB MopflaHOBHX KOJieij 57 Mop$H3M. B chjiy npefljiostehhh 5, KOJIBUO HBJineTCH noatowy cnemiajibhhm KopaaHOBUU KOJIBIIOM. 3AME4AHHE. OieBHflHO, ITO ecjm npohsbojibhoe KOJIBUO (/2 TBOpHeT TOHfleCTBeHHHM COOTHOUleHHHM TO B^ BunojiHHeTCH TOSTtecTBeHHo: J-(x,u,z)=O 06paraoe sepao, B aaflmthbhoii rpynne KCJIBIIG $ HQT ajiewentob nopajka 43., qio ecjih KOABIJC (5 yflobjieibophet (22) H He nue?i nopflflka 2 H 3 B aaahthbhoa rpynne, TO OHO sceraa nokajib- HO HHJIBnOTeHTHO. B camou aejie, BOSBUBM KOJiBqo ero ymhokeaaa /C H odpasyew KOJEB- ^. B KO^Bue K(±) see onepaiopa ymhoscehhfl 7?^ } ^re d odnoako^bito $?. ^ - cnetpiaabhoeftopflanosoko^bijo (5ea 3JienopnaKa 2 H 3 B aaflhtmbhoa rpynne. Kpowe Toro, HHfleicea 3. TenepB B KOJitu.e noakojituo ^ nopoaaaeich n odpa- 3 fa,..., fn. Ho Toraa B KOJiBi;e A"^ noflkojiiuo ^^ \e -frjftfj,,---, Rf-^ Ho ^"^ - cneunajibhoe^-hhjn,- HHaeKca 3 H nostowy no[2j HHJiBnoieHrao HHaeKca?7z. A aio, T3K K3K Tty^Hy = ^Xt/» MTO ecm ^ " npohsbejehhe H6 wehee qew TTZ, sjiemehtob HS ^", TO /^ = O IloaTouy KOJIIijo F HHjiBnoieHTHo HHfleKca He surae Men 772+ 1. Tew caiihm BOKajiiHaH HHJiBnoTeHTHOCTi JlHiepaiypa 1. H./lseKodcoH. CTpoeHae KOJIBU. HJI, 1961. B 2I.I.I966 r. 2. A.H.fflHpmoB. 0 HeKOTopbtx HeaccouaaTHBHHX HHjn>-KOJiBn.ax H aaredpan^eckhx ajiredpax. Mai.cd. 41 N»3 (1957), cip. 381-394.
58 K.A.IeBJiaKOB 3. A.A. Albert A structure theory for Jordan algebras. Ann. of Math. 4-8N 2 (1947), 546-567. 4. r.b.j(opo$eeb. Ilpuuep paspeuihmoro, HO He HHJii>noTeHTHoro am - TepHaiHBHoro KOJII)^a.-yMH, xy K»3 (I960), 147-150.