Anre6pa H norHKa, 9, Ns 6 (1970), 714-730.

519.48

OB ynOPHAOHEHHblX

n.H.CK/iM*oc

1 . B a e n e H H

COCTOHT H3 rpex HacxeS. B nepaofi nacxii noicaabiBaexca xe-

opeMa, yKaabiaaioinaa Heo6xonnMwe H AOCxaxoiHbie ycjioBHa ana xoro, HXO-

6bi peiiiexoHHO-ynopaflOHeHHafl Jiyna 6bHia CBo6oaHoii a6eneBott rpynnoii.

Bo Bxopoii 4acxH neKoxopbie peayjitxaTbi o /iHaeiiHO-ynopanoqeHHbix rpyn -

nax nepewocaxcH na nnHeHHO-ynopsifloHeHHbie nynw. B xpeTBefi Hacm H3y-

naexcn rpynna, accounnpoaaHHaa c jiHHeftHO-ynopanoHeHHoii nynoft.

PaSoxa BbinojiHanacfa aa xa^enpe Bbicmefl MaxeMaxHKH KniuHHescKoro

no.THTexHHnecKoro HHcxHxyra H B Macxuxyxe MaxeMaxHKH CO AH CCCP.

Asxop ciHxaex CBOHM aonroM BtipasHxt. 6/iaronapHocxfa H.M.Banyus,

non pyxoBoncxBOM Koxoporo 6bina BbinojiHena pa6oxa, a xajoxe oxiarona -

PHT yiacxHHKOB ceMHHapa "AnreSpa H .norHKa" oa pan noneoHbix .iaMeia-

H H H .

II. OcHOBHbie onpeaeneHHa

/lynoii HaabiBaexca anreSpa "?/=• < Or, • , /, N > Tuna -̂ 2,2,2> B

Koxopoii nna TIIOOWX SJ IOMBHTOB sc, y cnpaseajTHBiii c;ie,]\e C O O T H O , H MHI

2) (joy^/y =jc , y

3) j c / j c=y \z / .

/lyna y =• < Gt, •, /,\>HO3bi,'iaeT< a H C I C T K H H O ynopa .uo ' ioHHOf i nynoi'i

1) C? - 4acxnHHo ynopaaoneHHoe MHO/necxBo c O X H O L I J G H M - S M

O6 ynopHfloieHHHX nynax _ 715

2) Ana acex rpex onepaiiHH • , / ,~^ cnpaBeonHB aaKOH MOHOTOHHOC-

TH C o6jiaCTfaK> MOHOTOHHOCTH Cr , T.6. flJlH rtK>6bIX Cl H £ H npOH3BOrtfa-

Horo ce j

ct. ̂ & BneieT ca ^. c. & , CLC- ^ -&c ,

CCf. ^, C.S HJIH CtC ̂ &C sneneT Ct £, & .

ECJIH lacTHHHp ynopafloneHHoe MHOWGCTBO G asnaerca pemeTKoP, TO

nacTHiHO ynopaaoneHHaa nyna •§" = < G - , « » / , \ HaatiBaeTca pemeTOHHO-

ynopanoHeHHoii nynoft.

Ecnn HacTiiMHbiM nopHAOK B lacxHiHO ynopanoneHHoft nyne

TO nyna HaabisaeTca nHHef lHO ynopafloneHHOH.

3neM6HT CC. HaabmaeTCH nojio>KHTenfc.HbiM ( cTporo

ecnn cf. >^ (_ d > / ) , n oxpnuaTenfaHbiM, eonn CC ±~ / .

MHOKGCTBO no,no>KHTenfaHb[x sneMGHTOB lacTHHHO ynopHAOieHHoft nynbi

@ SyaeM o6o3HaHaTt> nepea £r . KaK noKasano B C 6 1 , nacTHHHbifi no -

paaoK za. B lacTHHHo ynopaaoneHHOH xiyne -§f onHosnaHHo onpenenaeTCH

MHO>KeCTBOM {P .

OonMHoxecTBO H lacTMHHO ynopaflOHSHHoro MHO>KecTBa G HasbiBa -

erca BbinyKiibiM, ec.nn H3 CL < 30 < O H Ct, Q G H cneayer jce /-/

BbinyKnofl nofljiynoft nynw y HasbisaeTca noanyna, asnfliomaaca Bbj -

nyKJIbIM nOflMHOJKSCTBOM B (j

Oxo6pa>KeHne jc — *-Gt3C HaaweaeTca nesoH TpancnauHeH c

cc H oSoaHanaeTca nepea L Q : L/a(^C) — a JC .

HCHO, HTO ^,-y aBnaerca noACTawoBKOH MHO>i<ecT[ia G , T.e.

OTo6pa:i:eHneM MHO>KecTBa G' na ce6a.

onpenenaerca npasaa TpaHcnnuna c noMomBW

nepea & '. 3C — , . .

HO, TaK»e HBnaexca noncTaHOBKOH.

Fpynna flTff, (^f) , nopo/KfleHnasi T p a H c a a u H H M H /^,~ , £(% j.insi nioooro

CCG- ^/ , HaabisaeTca rpynnof i , accouHHposanHOH c nynofl S/ C3H ,C4l.

FTonrTaHOBKa oc e. 'dTTt('Sf) naahiBaerca B i f y r p e n H e i i noacTaHOBKOi i , ec,nn

oc(/; = /.

Bee B i t y T p e m u i e no / i c ra i iooKH oopaayioT rpynny. O6o3naHHM 3Ty rpynny

•19 pea C/ .

L 3 ~2 , L43 , 'ITO rpynna Cf nopo/i\,aaeTca n

^Ly, z = e

716 n.H.CKJiH(j>oc

rionnyna H nynw V HaawsaeTca HHBapHaHTHoJl B y , ecnH CfH—

Mfai SyneM roBopnxib, ixo peuieTOHHO-ynopanpHeHHaa nyna 5/

aopaex ycnoBHio MHHHManbHOCTH ana sneMeHXOB, ecnH Ka*floe HenycToe MHO-

wecTBO nono>KHxenbHbix aneNEHTOB HS "?/ conepxHT MHHHManfaHbiJi aneMeHT.

3jieM6HTyD>/ pemeTOHHO-ynopanoHeHHOH nynu Hasbiaaexca axoMOM.ec-

Uenb sneMeHTOB a'<-...-< dn HaaoseM KOMnoaniiHOHHOH, ecnn ona He

HMeeT ynnoTHeHHH B ~§f , OTnHHHbix OT nee.

flse KOMnosHUHOHHbie uenn naawBaioTca noao6HbiMH, ecnH cosnanawT HX

Havana H KOHUM.0 .0KoMnoaHiiHOHHyra nenfa o <...< f)n HaaoseM cBepxKOMtioaHUHOHHoft, ec-

nH oaa coBnaaaex co sceMH en nono6HbiMH.

Ecnn Ci <....< CC.fi (1) asnaeTca KOMnoaHUHOHHOH (cBepXKOMnoanuH-

OHHOH) uenfaio, TO

(2)

TaiOKe aBJiaiOTCa KOMnOSHUHOHHblMH (cBepXKOMRO3HUHOHHbIMH)

AeHCTBHTenfaHO, ecnn XOTS 6bi oaua H3 ueneft (2) He aanaeTca KOMIIO -

3HUHOHHOH (CB6pXKOMnO3HUHOHHOH) , TO, paaaenHB HnH yKlHOJKHB Ha C C CO -

OTBeTCXByromeH CTOPOHM, nonynaeM, HTO H uenfa (1 ) He aanaexca KOMHOSH -

UHOHHOH ( CBepXKOMnOSHUHOHHOH).

3aM6XHTb, HXO ZUIB nlo6bIX 3neM6HXOB C£ , O H C. peilieXOHHO-

nynbi ^ HMeex MBCTO (ciA £)c= CtCA &C .

Ci H & pemeTOHHO-ynopanoHeHHOH nynbi y naabmaioTca

ecnH CC A o ==1 / .

=>, HTO ecnuc(A&= / H OC.AC— /, TOQ:A^C=/ .

Ecnn cf_ , • • -, cf-r, - nonapHO opToronanbHbie aneMeHTM peiueTOHHO-7 c

nynbi •§? , TO

a. v. . . v <zn= ar ...a.n. (3)

PaaeHCTBo (3) noxaabiBaexcfl KHnyKUHeii no n. .

PasjiHHHbie axoMbi pemexoiHO-ynopanoieHHOH nynbi ~& opToronanfaHbi

npyr npyry.

O6 ynopaaoieHHbix nynax 717

peuieTOHHo-ynopaflOHeHHbix iyn c ycnoBueu MHHHMajibHocTH

HMeeT M6CTO

/I EMMA. II e n b / -c /7<c . pz , r n e , o - a T O M ji y n H

'•& , y f l O B n e T B O p a r o m a H y c ji o B H ro M H H H -

M a n f a H O C T H a n a s j i e M e H T O B , a a n n e T c a

C B e p X K O M I T O S H U H O H H O f i U 6 I I b K>.

HOKASATE/IbCTBO. Uent> -f < p < pz aBnneTCfl KOMnoamuioHHOH ne-

), T3K K3K eCJIH

TO, paaaenHB noaxoaamyro noauent> uena (4) na y3 cnpasa, nonyiHM

H T3K K3K yO HBJIH6TCfl 3TOMOM, nOJlylHM i///2 ~ / IWH y

roro, ITO nyna & yaoBJierBOpaeT ycnoBHro MHHHManfaHocTH

, nroSaa ee KOMnosHuHOHHaa uenfa, nono6Haa uenii

4 < p

HM66T BHH

me o - axoM.

C^ H 73 HMeroTca use BOSMOJKHOCTH:

9^/3. 2)^ = ̂ .

flonycTHM, ITO BbinonHHexca zc ̂ O , xoraa

s

2 3 ,•JTO H8BO3MOKHO, TBK K3K H3 (6) HM66M SC <. £) H TOTfla 3C A/?

Ecm jc = n , TO ii3 Toro, HTO ueiib (5) KOMnosHUHOHHaa,

roBopHTfa, HTO onpefleneHo o , ecjin npoHsseaeHHa ,"i_

MHo?KHTeneft, paBHwx p , He aaBHcax OT pacnpeneneHHH CKoSoK.

C/lEflCTBHE 1 . E c n H - o - a T O M n ^ o o n p e n e n e -

H o , T O

/z C7)( '

•/< p

H B j i a e x c a c B e p x K O M n o s H U H O H H o f i u e n f a i o .

HOKASATii/lbCTBO IIPOBOHHM HHayKUHefl no n. . Una n,— 2 yTaep»-

cnpaaeanHBO. FIpennojioMHM, HTO yTsepxaeHHe cnpaBennnBO

H noicaxoM ee cnpaaeanHBocTb zuia^t= K "f~ / .

718 n.H.Cicnn(J>oc

fleftcTBHTenbHO, npn /ri-AC-f-/ iienb (7) npHHHMaeT ana:

<+-' (8)/</> < • - • </> .

npoH3BOJibHaa KOMnoaHUHOHHaa uenb, noflo6nafl uena (8 ) , HMeer BHU:

(9)l '

-aroM.

paCCyXAGHHHMH, K3K H BbllUB, yCTaHaBJIHBaeM,

p •Torna uenb. (9) 3anncwBaeTCH

PaaaenHB noflxoaaiiiyio noauenb. uenii ( 10) Ha /J cnpaaa,

Ilenb (11) B cany npeflnono>KeHHH aBJiHexca caepXROMnosHuHOHHOH, xorna-? • x-t 2

-, /P =P ,^-2 4**p^.,*yp-p "™ JC-frt-l = Pj£ =n <•£=/•)*• , r.e. uent. (8) aanaerca cBepxKOMnoanuHOHHOH, HTO sa-

m — 2 / > * " ' / rBepmaeT aoKaaarenfaCTBO cnencxBHS 1.

uenb>

p\ < (p ^ /;/? <. p .

Ilenb ( 12) SBJiaeTca KOMnosHUHOHHoii uenbK). VMHCOKHB c^iesa aa p _, nojiyqa-

6M, HTO

/J ((n \ )a) =^ o HJIH

T.e. , x /= //^ = p~ .

Mcxona as BbiuienoKaaaHHoro^-/

C/IEflCTBME 2 . E C H H p - B T O M , T o p < /< / o a B-

n a e r c a c s e p x R O M n o s H U H O H H o f t u e n b i o .

C^EflCTBHE 3 . E c n H / 7 - a T O M , TO (p~/)Z<p ~' < i

H B j i a e r c a c B e p x K O M n o s H U H O H H O f t u e n b r o .

C/1EACTBHE 4. E C J T H / > - a x o M *•(£>*'}"' o n p e n e -

j j e H O , T O u e n b ( /O ) < (P~f) <...<iO~f< / a s n a e T -

c a C B e p x K O M n o s H U H O H H O H .

TEOPEMA 1. P e u i e T O H H O - y n o p a n o i e H H a a

n y n a y x o r z i a H T o n t . K O r o r a a a s j i s e T -

_ O6 ynopggoHeHHbix nynax _ 719

C H n p H M H M n p O H 3 B 6 f l e H H e M J I H H 6 H H O -

y n o p a f l o n e H H b i x u H K n H H e c K H x r p y n n , K o -

r n a o H a y n o B j i e T B o p a e r y c n o B H K> M H -

H H M a n b H O C T H f l J I H 3 J i e M 6 H T O B .

flOKASATE/lbCTBO. VTBepjuaeHiie "TOjibKO roraa" - oneBHUHO, nos -

TOMy npeano.no>KHM, HTO ^ - pemeTOHHO-ynopaaoHeHHafl nyna c ycjioBHOM

MHHHManbHOCTH AJIfl 3JISMeHTOB.

AJIH nroSoro CL G. & , CL > / , MHOJKecTBofjce ^f\f^^c^cT\T

MHHHMajlfaHfalH SJieMeHT Q , KOTOpblH, OieBHflHO, 6yfl6T 3TOMOM. EcjIH C2f =

= n \ > / , TO Mbl CHOB3 nOnyHHM 8TOM D , yflOBJI6TBOpHlOmHH yCJJO-

BHIO / < p ^ aff •

ECHU c£=* p \f >• •/ , TO MM 6yfl6M nponojiKarb STOT npouecc no

Tex nop, noKa ne aoftfleM ao HeKOToporo <2f •=» f>^ N Of _. = i . Tax KBK

OC>Gl1 >C(Z > . . . , TO ycnoBHe MKHHManbHOCTH nna sneMeHTOB o6e -

cnenHsaeT cymecTBOBaHHe TaKoro K. . Toraa ct =-n (n ... (a . p). . .) •

rne p ^ , £=/, 2, ...,/< } - aTOMw.

O6o3HanHM lepea < p >noanyny nynbi ^ , nopo*AeHHyio aTOMOM p ,

H AOKa^ceM, HTO -<jC>> asnaeTca accouaaTHBHoii nopManfaHoft nofljiynoii

- aTOM, TO nerKO noKaaaTb, HTO

KOMnoanuHOHHOH

noflxoaamyKi noauenB uena ( 13) Ha /j cnesa H3 2 '

neMMy, nonyHHM, qroyO ^D*O —p , HO STO paaaocHnbHO TOMy,

~p-p=-p , T.e. /O onpeneneno. npeanonojKHM, HTO D onpeaene-

HO, H AOKaJKeM, HTO onpenejieHO H

PaccMarpHBaeM CHCTeMy HS n. ueneii, KOTOpbie, KaK nerKO

HHiOTCa KOMnOSHIIHOHHblMH:

n n

Paaae/tHB Ka>Kayro KOMnoaHuaoHHyio uenb CHCTeMbi ( 14) cnpaBa COOT -

BGTCTBeHHO H3 p ? ___ ? /? , IIOJiyHHM CHCTBMy KOMnOSHUHOHHblX IieiieH:

.,1<p . (15)

Flo npeanono>KeHHio H B cwiy cneacTBHa 1, nonyHHM CHCTeMy paaeHCTB:

-7 a. n-s 2 2 n—/ n r^

720

Aanee paccMaxpHBaeM cncx6My uenefi, Koxopbie, KBK ner-KO Bunexfa,

T3KJKB HBnfllOTCa KOMnO3HUHOHHbIMH:

ft j n-i 2 /»-/ 3 2 /z-/ 2 n. M7p,p p^p p,...,pp <p -p . U'

Kamnyio uent. cncxeMbi (17) cooxBexcxBeHHO uayO . f p

cnpaBa, nonyHHM cacxeiviy KOMnoaHUHOHHbix ueneft:

Flo npennojiojKeHHK) H B cany cneACTBasi 1 nonynaeM cncxeMy paseHCTB:

n-i 2 n-y 2 1—2 3 2 /?—•/ rz M Q >

Toraa nonyneHHbie paseHCTBa (16) H (19) onpe aensrox O

paccy>KfleHHHMH H B CKny cnencxBHa 4 nonynaeivi, HXO-/ n(_p ) onpeneneHw AJIH nro6oro HaxypanbHoro Hucna /t.

, HXO ansi naHHbix HarypanifaHbix queen ATT H /"Z onpeneneno

n ) p , ecnH see npoHSBefleHHH, conep>Kafflne m. paayC7~ H /-f_ paa

,O B npOHSBOJlbHOM IIOpSIflKe H npOHSBOJIfaHOH paCCXaHOBKe CKO6OK, paBHbl

^ o n—rrtCODOH H, KaK nerKO Bnaext,, xorna pasHbi p

, HXO Ana nio6bix HarypanbHbix HHCSJI /T^ H /T. onpeneneHo

STO HHayxuHeii no rri

EcnH/rt=/ , xo yxBepwneHHe oneBHflHO, X.K. p ~^p =Pp ' = /•

HpeflnonoKHM, 1x0^/3 ^ ,3 onpeneneno. Toraa ^ /j )rrt< Cp ) p<

<(p ) p . CornacHO npeflnonoKeHnw (p~l)'~n~p—(p~'')rn~ . TaK KaK

CBepxKOMnoaHUHOHHofl , To(p~')rrip = ( £3~ )

(1 npeanojiojKeHHe) , HTO ansi/t — / onpeneneno npoasBe-

n ~* npw jiroSoM rn. •

HXO xorna onpenejieHO n npOHSBeaeHHeAo ^ /? npa

fn. . floKa>KeM 3xo KHflyKUneii no/~n. . flna rrt,— -/ HMeeM (" o j a

•6-')p™<;p-')pn'<pn- .n <• — , ~ /*— ~^Ho nepsoMy npeanonoweHHio^> ^^ =p , x .e . /> -<(

n-2, /?-/ /z -j n n-/'-HO X.K. uenb/2 <^ /? ^/^ cBepxoMrrosHUHOHHaH, xo ( p ) p = r>

FIpennonowHM xenepfa (FI npeflno,no>i<eHne) , 4x0^/3^ n HMeex Mec-

xo nnfl naHHoro n. . Torna

3necb ( p !) p p no nepsoMy npennono»ceHHK),

O6 ynopanoneHHbix nynax 721

f _ l)m~pn — p /nf/+/"J no BTOpoiny npennono»eHHio, TO ecxt>

., - rn—ft-n —Ti+n — m+rt.+ fHo uenfa o P- P asjiaexca csepxKOMno -

3HUHOHHOH, cjienoBaxenbHO, /) = (n *) n , HTO rpe6oBanocfa floicaaaxb.

HxaK,<o> aBJiaerca accomtaTHBHoi! nojxnynoH c pasnuinbiMH crene -

, TO ecTb 6ecKOHeiHoft umtnuiecKofl rpynnoft.

IlycTfa ct> / n nycTb

a-^r^...^.,^)...) (2D

paajioweHne aneMenra C?

(22)-apyroe aroMnoe pasJio>KeHHe sneMeHra <3T , TO B paanoKBHHe (22)

T9 )K6 aTOMBI, HTO H B paSJIO^KSHHe (21 ) .

fleflcTBHrejiBHO, p • = n . ana HeicoToporo i, , xax KBK BCJIH 6bi/?. ^£y?.' « £ . ' £ ^t C

Ana Ka>Knoro t,— /}2,...} K > T°/O • A Q—f • Ho STO HesoaMOJKHO, raic

KaKn. ^ Q; H, cjieflOBaTenfaHO,/? . A.(Z=n -I l^ ' tz ' 6r

* * , . . .

B pa3no>KeHHH (21) aneweHra O ,̂ CHHraeM HHCHO BXO-

/j

HycTfa p BxonHT /T. paa. Toraa nerico aaivieTHTfa, HTOCZ ^/5

jieMSHTa <3J- HMSBM paanojKSHHe Gt— D fGCf , rue <2"

CC, > -f , TO HJIH sjieMOHTa <3?r cymecTByex axoMHoe

B paaJioxeHHH (23) CHHTaeM HHCJIO Bxo«fle«Hft axoMa /? . HycTfa O^ axo-

n* paa. Toraa, onesHflHO, <2 =p 9 • Of. , rae Ct ^ y ./7 J_

o6paaoM, Q:= ^ ' — *1' ** ' >7 V f ~2J /-/ ^2 2

ECJIH <X > / , TO MH SyaeM npoaojixaxfa axox npouecc no xex nop,/

noKa He aofiaeivi ao HeKoroporo CX, , B axoMHOe paanoaceHHe jcoxoporo y«e

H6 BXOflHT aTOM O . TaK KaK <2 >C^.>Q: >... , TO yC/IOBHe MHHHMajIbHOC-

TH oSecneHHBaeT cyiuecTBOBaHHe TaKoro ^~

HiaK, CL=n^r cc.' , rae rnf=nf +n'+...+/? «/>r «e BXOHHX B axoM-

Hoe paano>KeHHe aneMeHxa Oi .

EcjiH<2. > / , TO <2 /= n rn' ClK > rae ce. = / H /? He BXOAHX B

axoMHOe

722

Ct > / , TO MW 6yneM npoaonwarfa STOT npouecc ao Tex nop,/

noKa He nofineM no neicoToporocj: = / .S

Tax naK.cZ > CC '> CC.^. >.. , . TO ycnoBHe MKHHMajibHOCTH ana ane -K K1

MSHTOB o6ecneiHBaeT cyraecTBoaaHHe xaKoro xc_ .

TaKHM o6pa3OM, ana sneMeHra d. nonyiHM

TaK KBK paanojKeHHe (24)^ •J^/J- • ecnn t ^t=J , TO B CHJiy opToro -J

paanHHHwx aTOMOB nonyiaeM, ITO B paajKmeHHH (24) CKOSKH MO-

WHO onycKaTb, TO ecrb

a -/>7r/°r*' • • />?* -/c^ • (25)flanee noKa>neM, HTO</>> asnaexca HHsapnaHTHofi nonnynofi nynw y

fleflcTBHTenbHo, ana jiK>6bix 30, y '& "§/ H o HMeeM: 7~ (a )

4r.y (*"> > / H & (/&>' ' ecnK rL>° '' H

Tenepfa, yiHTbrnaa paaeHCTBO (25) H aHanoraiHoe paseHCTBo

B , - , • - , •.. / •_/ t / * ' tf f

fleftCTBHTejTBHO, 6CJ1H &l>l<D->/'~><Oj)...)/3-_ > = C 7= / , TO, H8

o6lHHOCTH, MOJK8M CHHTaTfaC1 > /• To VHB C=/O *** 0 * /3**'~f ' rfle/^7 / jg "•/ f— f

- nna/^ €•<. i, — / HC=/0- flna seKOTOporo /z > Q •

C npyroft GTOpoHbi.yO-^/^, ana /^ ^"^ t~/ • Torna, B cany TOFO.HTO

pasnHHHbie aroMW opToroHanfaHbi flpyr apyry, HMeeM n^Ar}*' /O- *~*=-/ -•' i, ' T '" / &~r '

npoTHBOpeiHe. Teopewa

§ 2.

Ooa n. -KBaaacTeneHfaK) ane.MeHTa Ci nynw y 6yaeM noHHMaTB sne-

. . . cc,l c neKOTOpbiM pacnpenene HH9M CKo6oK.

ByneM o6o3HanaTb n. - KBasHCTenenb sneMeHTa CL nynw ^ nepea/t

CL

/lyna y HaabiaaeTCH nynoft 6ea KpyieHHa, ecnH flna nioSoro

vis cc- ==- -f cnenyex ct, = / .

/lyna, KOTOpyro MO>KHO cnenaTfa nHHeftHO-ynopanoneHHofi,

ynopanoHHBaeMOfl.

Ha onpeaejieHHa jiHHeHHO-ynopnno<ieHHOfi Jiynu cneayex

O6 ynopsmoHSHHbix nynax 723

nPEflnO>KEHHE I . Y n o p a n o H H B a e M a H n y n a

e c x f a n y n a 6 e 3 K p y i e H H f l .

IlycTb •& = /7/7 - nojjHoe npaMoeoCHblX Jiyn /7 , MHOWeCTBO HHfleKCOB / oC \X BHOJIHe

oC *• JBsefleM B €f nopanox, HaawBaeMMft jieKCHKorpa4>HHecKHM, cneayiomHM

o6paaoM; C(.=-(... ,Ot ' . ..) H3 •§/ CHHxaeM nonojKHTenfaHWM, ecnHC^.^/

B fl. H<2 r~= / B tfa flnH Bcex/3<<x • npn STOM nopaAKe •& Syner nn~OC ^ p sHeJtHO-ynopflnoieHHoft nynoft, H, sHaiKT, cnpaBeanHao

nPEflJlO^(EHHE 2 . I l O j i H o e n p a M o e n p o H a s e -

n e H H e y n o p a n o i t i B a e M b i x n y n y n o p a n o -

H H B a.e M o.

<J>aKTOp-nyny & / H nHHeftHO-ynopafloneHHOfi nynw ^ no BbinyKnoii HH-

BapnaHTHofi noanyne // MOWHO caejiaTfa nHHeflHO-ynopafloneHHofl Jiynoil cxne-

ecrecTBeHHbiM cnocoSoM: nonoxHM cC/-f< -@H B ^"/// , ecnn H

ecnH cc'< & B ^ nna

FIpHBeneHHoe onpeaeneHHe SKBHsaneHTHo

y3^"?/ / H) ecTb o6paa &'(•§') npa roMOMOp4>H3Me ^ Ha

Tax BBeneHHwft nopunoK na <}>aKTOp-jiyne HaabisaeTca

FIpH HflnyuapoBaHHOM nopaflKe BbinyKJiWM nonnynaM HS & , coflep«a -

UIHM // , COOTB9TCTByK)T BbinyKflbie HOflJiynbl H3 & /// , H o6paTHO.

FoMOMOp4)H3M </" nHHeftHo-ynopaflOHeHHOft nynbi ~§f B nHHeftao-ynopa -

noieHHyro nyny & HasbiaaeTca y -roMOMOpi})H3MDM, ecnH HS a -^ t? B ̂

cjienyer <£>(&) ̂ <?(.&) B .£" nnn, ITO paBHOcanfaHO,

3TOM <£>(-) = H V(p()= & < TO V H33bIBaeT -

CS U -3nHMOp(|)H3MOM.

H "/" ecTb y -SUHMOP^HSM, TO <P HasbiBaerca u -HSOMOP-

, a nynbi H

TEOPEMA 2 . H a B a p n a H T H a a n o a n y n a / / J I H -

H e H H O - y n o p H H O i e H H o f i n y n H ^ r o r a a H

T O J i f a K o r o r n a a a j i a e x c a a n p O M H G K O -

T o p o r o ^ - r o M O M O p t ^ H S M a H U H y - a n n -

M o p 4 ) H 3 M a , K o r n a o n a B b i n y x n a H . E C J I H

(f a s j i a e r c a y - a n H M O p 4 > H 3 M O M ^ H a •& c

a n p O M / / . T O & 'H 3* & .

OyCTb H - aapO TJ -rOMOMOp(|)H3Ma V JIHHeHHO-ynOpflflOHSHHOH Jly -

nw -^ . Torna H3 a«JC« ^ H Ct.gG./-/ cneayer <f(l)= <P(CL)

724

— / , OTKyaa <?(•/) = iPCx) H, anaHHT.jce// , TO ecTb H

Bbinyicnaa.

ECJIH // - HHBapHaHTnaa H Bbinyicnaa noflnyna nynbi ^ , TO TOMO -

MOP$H3M -§f HB & /H C HHAylHIpOBaBHblM HOpaflKOM 6yfl6T Tpe6y6MbIM.

ECJIH <P ynosneTBOpaeT ycnoBHSM BTOporo npeanosceHHa TeopeMbi,

TO COOTBBTCTBHe O' - *-*?'{<}) SyflBT IJ -H3OMOpt{)HaMOM ~§f H •g' /// .

nPEJUlOKEHHE 3 . E c n n ( J i a K T O p - n y n a ^/// n y -

n w v n o H H B a p a a H T H o i i n o a n y n e // n H -

H e H H O - y n o p f l f l o n e H a H / / a o n y c K a e T J I H -

H e f t H bi f t n o p a / i O K , y c T o f t H H B b i i i O T H O C H -

T e n f a H O B a y T p e H H H X n o a c T a H O B O K n y -

n b i ^ . T O B n y n e M O « H O B B G C T H T a K o f t

n H H e t t H b i f t n o p a n o K , n p n K O T O P O M H H -

n y i i H p O B a H H b i e n o p a n K H H a A / H ^ / / / 6 y -

a y T c o B n a n a T b c a a n a K H b i M H n a H H X

H H 6 y n e T B b i n y K n o i i B ^

fleftcTBHTenbHo, nK>6ofl anewteHT O'e^' npencTaBHM B BHAS CZ/4. , rne

ft,€s. // , a c£ - npencTaBHTenb> B CMSJKHOM Knacce Cf./-/ .

nio6bix nsyx npencTaBHTenefi CJL H Of CTHSHM c? <• Cf , GCJIH H' a f £t

ecjia d H < Q H B $ /M •I 2i

aannbift nopanoK cneayromHM o6pasoM: ana nioSbix ane -

tcfz/z2e. $ CHHTaeM or Af < QT A , ecnH &r< <Z, • SCHH

-to CZf/ir <af/zz , .ecnHy£r</4^ B // .

nopaflOK aanaercH jiHHefiHbiM, H // aenaeTCH BbinyKnoft

noanynoft OTHocHxenbHo nocTpoeHHoro nopaflxa.

C/IEflCTBHE 1 . K a a c a a a n H H e f t H o - y n o p f l n o -

H e H H a a n y n a a s n a e T c a ^ - a n H M O p c ^ H b i M

o 6 p a a o M n e K O T O p o f t n H H e f t H o - y n o p a a o -

l e H H o f t c B o S o f l H o f l n y n M .

HycTb y - nHHeHHO-ynopanoneHHafl nyna. Ona KaK a6cTpaKTHaa nyna

H3OMOp(|)Ha <j>aKTOp-nyne 5y// HeKOTOpoft cBo6oflHoH nynw & ,

B

nyna, roraa H H nHHeftHo ynopsmoHeHHaa.

HopaaoK B // ycTOflnHB OTHOCHTenfaHO BHyTpenHnx noACTaaoBOK nynw

, H Tenepb, npHMeHaa npeanoMceHHe 3, aaaepuiaeM AOKaaaTenbCTBO.

TEOPEMA 3 . f l n a B c a K o f t j i H H e f i H O y n o p a -

e H H o f i n y n w c n e a y i o i n H e n s a C B O H -

_ O6 ynopaflOHeHHbix nynax _ 725

c T B a s K B H B a n e H T H b i M e JK f l y c o S o i i :

!)• ji n H e f t H o y n o p s n o i e H H a a n y n a y

He H M e e T c o S c T B e H H b i x B M n y K n bi x n o a-

n y n;

2) n n a n to 6 w xjc y / , .y > -f ~,JC, y e. & c y m e c T -

B y r o T H a r y p a n b H w e H H c Ji a /~n , /Z T a K H e,

i T o jc™?- iJ n yrn>jc. {JC ,y - H e K o r o p b i e K B a a H -

c r e n e H H s n e M S H T O B JC. n c o T B e r c T B e H -

H o y ).

Cnenyiomaa neMMa Heo6xoaHMa ona npoBeflenMa flOKaaarenbCTBa Tonb-

KO HTO ci})opMynHpOBaHHoft reopeMbi.

/1EMMA. E c n H /?= <X> - M O H o r e H H a a n o n -

n y n a j i H H e H n o y n o p a n o n e H H o i i r t y n b i

f y , T O M H O > K e C T B O 3 n e M 6 H T O B /•/= / ̂ C. & /

Ct^U< S ', Ct , O G. /7 \a B b i n y K n o f t

n o n j i y n o f i B ^ " .

flOKASATE^bCTBO. Hycrb Ctf<l/r<£f H Q^ <J^ < Z^ ,a

rpynnoHflOM.

FIpoBepHM, HTO peiueHHH ypaBHSHHfl CZ5C= O H l/CC—u ; ct. DG/f, npn~

K

/lerxo npoBepaerca, HTO pemeHHa ypaBHeHHfl ct3C = D & IJCX =• O

npaabix H neBwx TpaHcjismufi aanHCbiBaioxca B BHfle:jc = ^ , C&) H

PaccMarpHBaeM

Toraa / -c Z < cf-

. - -Tax KaK z^-Li C^ H OHeBHnHO BwnonHaeTca La(-t)-£ L

i -, TO, cneaoBarenbHo, L

-^ -/ ,_/ -/ ./, # =/ C^n 1<Ly(.yz)*l<,,ty<£ci(.-t') , a "

726 n.H.CKJiH<J>oc

~ (yr) H/r ta)* L~v Ca)<Ll/Ct/r)</' r° 6~

~

_f ,H Ha 3TOT pas Z—£iv CtS ) H ̂

-*

Bbiiue cnynaa nosBonawT aaKnioiHTfa, HTO ecjiH C£,

H JC yAoaneTBOpaeT ypaBHeHHio CtX—& , TO JCG. // .

TOHHO rax »e MOMHO noKasaTfa, HTO ecjiH CCt f) &// H _^

nexBopjieT ypaBHeHHK) i/GC = o , TO ̂ S /*^ . CnenosaTenBHO, X/ - noany -

na H, oneBHflHO, BbinyKna. /leMMa aoKaaana.

AOKA3ATE/lbCTBO reopeMM.

1) =^-2). fleiicTBHTenfaHO, nycrb jiHHenHO ynopflno^eHHaa nyna *y

6ea HCTHHHBIX BbinyKnbix noflnyn.

ECHH /7=^jC'> - MOHoreHHaa nonnyna nHHeftHO yrtopanoHeHHOH nynw

•§f , TO MHOWeCTBO 3JieMeHTOB //= { r^e. $ \ < IJ4- O }'&f&€/?] HO neMMB

aBJiaeTCfl Bbinyicjioft nofljrynoii H, cneaoBaTenfaHO, // coBnaaaeT c y .

A Tenepb, ynHTbisafl nocTpoenHe noaJiynw // , aaioiicAmeM, HTO 2)

cnpasennHBO.

2) — ̂ - i) _ fleficTBHTenbHO, nycTt. nna yiioSbix JC ̂ •/ , is y / cyme -

CTByioT HaTypanfaHbie incna /n, /z. TaKHe, HTO jc ?• i/1 K^ y-JC (.X-rtS -

HeKOTOpbre KBasHCTeneHH aJieMSHTOB jc H Z / ), H npennonoKHM, HTO cy -

mecTByeT HCTHHHaa BwnyKnaa notuiyna //

cymecTByer sneMeHT jC-^-f ,JC.€-y VIJC.G //.

ECJIH Tenepfa y e. f-/ , TO AJIB nioSoro HaTypanbHoro HHCJia ATT./TZ

o cymecTBOBaHHH HCTHHHHX BbinyKjibix nofljjyn HeaepHO.

TeopeMa AOKaaaHa.

§ 3

FlycTb 0TC (&) - rpynna, accounnpoBaHHaa c jiynofi •§/ . ECJIH & HB-

jiaeTca rpynnofi, TO &&%.('&) H30MOp<}>Ha £f

Tenepfa y - Jiyna, a 9Tfl(_ty- aSejiesa rpynna. TorAa

, (jc) AJia jno6wx ce/ &,i

O6 ynopsfloieuHbix Jiynax 727

KOMMyraTHBHOMy H accoHHaTHBHOMy aaxoHaM B

nPEZUIO>KEHHE 4 . &rt(&) K O M M y r a r H B H a x o r a a

H T O J i f a x o T o r a a , K o r a a & - a 5 e n e B a

r p y n n a .

Ha npenjioxeHHH 4 cueAyer, «ITO ecna y asnaerca KOMMyraTHBHoii

nynoft, HO He rpynnofi, TO &&%(&) - HeKOMMyraTHBHaa rpynna.

Fpynna •£/ HaabiBaercfl npaBoynopflnoHHsaeMoii, ecnn B MHOwecTBe ee

ajieMBHTOB MO>KHO onpefleJiHTb oTHouieHHe jiHHeftHoro nopaflKa,

cnpaaa OTHOCHTenfaHO rpynnoBoro

AHanorniHO MOJKBT 6biTfa onpeneneHa neBoynopafloiHBaeMaa rpynna.

IlocpeflCTBOM oTo6pa*8HHH 3C — *-jC rpynnbi §f aa ce6a MOJKHO ne -

peftrn OT jieBoro nopaflKa K npaBOMy.

riycTt> renept. ^ - JIKHSHHO ynopflaoneHHaa nyna, xorna cnpaaeflnHBo

nPEJinO>KEHHE 5 . 91fl(_3) n o n y c x a e T n p a a b i i i

n o p a n o K.

flOKASATE/lbCTBO. /lyny "̂ snojine ynopaaoqHM, T.e. sneMeHTbi HS

-^ CHHTaeM npoHyMepoBaHHWMH nopanKOBMMH

Ten.ept, Ha rpynne fflCt (_ ^f) onpeaenaeM OTHouieHHe nopaaica (<) : ane-

CfC. tffLC'Q) CHHxaeM nonojKHTenbHbiM, ecnH H ronfaKb ecnHQ'^c'e c ^

ana HeKOTOporo oc K ty(jc-) —JC^ ana Bcex/fl<oc B (1).

Ecna &,h.e-0VZ.t&) ^ > £, . /&">£, ( &• - eanHHua rpynnu

rorna qh, > &, .(7

, nycTfa -3C, - nepswii ajieMOHT HS (1), ua

C ) > JC H nycxt, ^c_ - nepsbifi aneMenT H3 (1), na KO-

neiScTByer /i(jc ') >JC .

He orpaHHHusaa oSmHocTH paccy»caeHHft, MoweM

V < <sC

= A

2) oC •* J& • B STOM cnyiae Q(3C, ) >• ^C _, ana Hexoroporo oc w.

= JC., ana scex \ «

ana acex ?< f3

728 n.H.CKJiH(t>oc

Toraa /T. — . »

yTBepwflaeTca, ^ro

() • npHifeM JCj. > JC

Tenepb, ecnn 6tr /L(JC^ < JC^ , TO /I ' ( fy, (jCg,)) < A'^-X^c )~

* . . . ,C- oc1) H 2) noKaawBaioT, HTO MHOKSCTBO nonoxHTenfaHbix sneMenTOB 06

paayeT nonyrpynny.

HTBK, nonyrpynna nonojKHTejifaHbix aneMeHTOB onpenenaex B rpynne

npaawft nopanoK cornacHo npaBHjiy Or >/i. Torna H Tonfaico

Korna Gf./i~ > £- ( '5' H A - npoH3Bont>Hbie sneweHTH na &7K, (&) ).

FlycTfa CHoaa ^ - nHHeftno ynopaaoneHHaa nyna, H snojiHe ynopano -

HMM ee cnenyromHM oSpaaoM:

j s,r jr (2)T . *-*-* > • " • 1 •-*-_*> » • • « ! \ * * /

rne / -. enHHHiia nHHeflHo ynopHaoneHHoK

Onpenenaa npasbift nopaaoK Ha rpynne 1?tt(Q) TOHHO ran, xaK STO 6u-

no cnenaHO Bbiuie, nonyiaeM, HTO rpynna BHyxpeHHHx noncraHOBOK Cf nw. -

ynopanoieHHoft nynti €f aanHerca BbinyKnoft nonrpynnoii B rpynne

nycrt /^e. ff , T.e./?(fJ=z / , H

ECJIH 7? - npoH3BonfaHbift snevieHT rpynnu ^/2T^"^ H£--^a?</7 i

na cornacHO onpeneneHHio npasoro nopaaKa oneflye.T g,< ''

3x0- BHaiHT, HTO ana nioSoro /g <oc HMeeM n g£~

, TO Torp.ag?(f) = / H BbinyicnocTb nonrpynnbt

rpynne flft (•&) AOKaaaHa.

FlycTb Teneptoc= / , x.e. ^ se'fa) > * HJ1H

npoTHBopeiHe.

FIonyieHHoe npOTHBOpenne noKaabmaeT, ixooC^^/ HGBOSMOHCHO.

J7 flBnaexca noarpynnoft rpynnbi &<T£-(&) HO, BooSme rosopa, He

HHBapaaHTHoft.

rionrpynnoft 3 nopo*AaeM &&&£(•&} HHBapHaHTHyw noarpynny Cf

- (fcaKTOp-rpynny rpynnbi fflZ(4/) no HHBapa-/~^

aHTHoft noflrpynne 3 .

aneweHT rpynnu <&%(-&)/ Zf npeacTasnaeTCH B

j j o i r x j C n i q a f l O H X H B H d B B H H g o d o x o x a H

o i l g 11 u A. it flOHHetoUBdouA' o H H 3 H H ir e u

- X i r - d o x x B ( { > B H < j > d o i N o e H - / 2 jC/(&)3£€ B u u A" d a

B B H H e h o t T B d o u X o H H e H H i/ > VWHdOHX

oxoew

OHjjaiilllf BH C. l1JlX.lt HOHBetOWBdOuX OHySHHl/ WOFMeH<j>dOF«

BOX8BITBB ^ 3 3 £> -t - 7?

BH xoDBdoii giqH{f8HHi: BBttotBiretreduo 'BiinXdaXirputroii ixoe

a / ̂ ( = 7 -x-x • S - a = jp 7 f 7=

S ~ -*j f -*s ., -^ f^f j—

XKaweue jfogoiif — ̂ ^H S"3^7 yniroa 'eireu WOWBO g

VC / (/•)•}?££ -3£:r7 xaoa Bim S" "3 g* ̂ J g3* 7)

/-

~ "a'x ' /=or

" / g

ox l $ ' & r / t S"^X7 C7 Hiroe '-N/ T

• s-^ S"« -t:nwBaxoifoao Hwnmc*x;ireifo xaeDBirgo oaxoascoHW oxg

a

a fi^f Hiroe oxii/ox H

'̂ y^/T H ' /T=X7'V oi ' /=. af" Hiroe oxfc 'ixaxewee osjajj1^^ -*x '̂ -^^

' -^ 9H

a A -5= or Hiroe OH«HTOX H Hiroa ' /7 ^^ G^T wesxHho :woeBdgo

("50 BMUBdou aHHamoHXo BoxajCdnnXDHH yC* / ^"72^6 euu-^dj g

XBUAU xHHHehoWBdouX go

730 n.H.CiuiH<t>oc

J l H T e p a T y p a

1. J1.4>yKC, HacTHHHO ynopanoieimbie anre6pannecKHe cHCxeMbi, M.,'Map", 1965 r.

2. A.H.KOKOPHH, ynopsnoHHuaeMbie rpynnw, Hfy, 1966 r.

3. R.BRUCK, A survey of binary systems, Berlin-Gottingen-Hei -delberg,1958.

4. B.fl;BE7IOyCOB, OCHOBH TeopHH KBaaHrpynn M nyn, M., "HayKa",1967 r.

5. D.ZELINSKY, On ordered loops, Amer.j.math.,70(19^8).

6. B.B.PUOKKOB, lO.A.CEHHBAHOB, HacTHHHO ynopaaoieHHwe ny -HEI, MareMaTHiecKHe HccJieaoBaHHH, KHUIHHGB, PHO AH MGCP, 2, N? 4,(1988).

7. n.M.CK/IH4>OC, Marepnajibi noKJianoB yi HayiHO-TexHHiecKOft KOH-<J>epeHiiHH KTIH, KHiiiHHeB, 1970.

28 Hiona 1970 r.