Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea

IAO?AA*OE Aiaoie?e Iao?iae/

OAeE 512.553+512.554

OAEOI?ECAOe?ss ? AeIIIAITHAAI?NOUe

A AEAAA?AO E? OA ANIOe?AOEAIEO AEAAA?AO

01.01.06 — aeaaa?a ? oai??y /enae

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

aeieoi?a o?ceei-iaoaiaoe/ieo iaoe

Ee?a-1999

Aeena?oaoe??th ? ?oeiien

?iaioa aeeiiaia a Ee?anueeiio oi?aa?neoao? ?iai? Oa?ana Oaa/aiea

Io?oe?ei? iiiiaioe:

?ssAOO?I TH??e Ieoaeeiae/, aeieoi? o?ceei-iaoaiaoe/ieo iaoe, aeaaeai?e
AI Iieaeiae, i?ia?aeiee iaoeiaee ni?a?ia?oiee ?inoeoooo iaoaiaoeee AI
Iieaeiae, i.Eeoei?a

NENAE ss?ineaa I?ieiiiae/, aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ia?aeiee
iaoeiaee ni?a?ia?oiee ?inoeoooo iaoaiaoeee IAI Oe?a?ie, i.Ee?a.

OIAEUeE?I Aeueo?aae Eueaiae/, aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?
eaoaae?e aeui? aeaaa?e Iineianueeiai aea?aeaaiiai oi?aa?neoaoo ?i.
I.A.Eiiiiiniaa, i.Iineaa

I?ia?aeia onoaiiaa:Euea?anueeee aea?aeaaiee oi?aa?neoao ?i. ?.O?aiea,
i.Euea?a

Caoeno a?aeaoaeaoueny “31” na?iiy 1999 ?. i 14 aiaeei? ia
can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae 26.001.18 i?e Ee?anueeiio
oi?aa?neoao? ?iai? Oa?ana Oaa/aiea ca aae?anith 252127, i. Ee?a-127,
i?iniaeo aeaae. Aeooeiaa, 6, Ee?anueeee oi?aa?neoao ?i. O.Oaa/aiea,
iaoai?ei-iaoaiaoe/iee oaeoeueoao.

C aeena?oaoe??th iiaeia iciaeiieoeny a a?ae?ioaoe? Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea (aoe. Aieiaeeie?nueea, 58).

Aaoi?aoa?ao ?ic?neaii “30” eeiiy 1999 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee ___________________ A.A.Ee?e/aiei

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Aeena?oaoe?eia ?iaioa a?aeiineoueny aei oiai
iai?yieo a aeaaa??, oaiaoeea ? iaoiaee yeiai ca’ycai? c ianooiiei
ieoaiiyi: ye? aeanoeaino? ia? aeaaa?a E? (aai anioe?aoeaia aeaaa?a) L
iaae anioe?aoeaii-eiiooaoeaiei e?eueoeai R, yea ?iceeaaea?oueny a noio
L=A+B aeaio nai?o i?aeaeaaa? A ? B c aeayeeie a?aeiieie aeanoeainoyie?
Oaeee ?iceeaae ii aiaeia?? c oai???? a?oi aoaeaii iaceaaoe
oaeoi?ecaoe??th aeaaa?e L (iaaaaea?ii, ui i?e aea/aii? a?oi c
oaeoi?ecaoe??th ciaoiaeyoue aeanoeaino? a?oie G=AB, ciath/e aeanoeaino?
aeanieo i?aea?oi A ? A c G). Iai?yi a oai??? aeaaa? E?, iia’ycaiee c
aea/aiiyi noi aeaaa? E? c oeie /e ?ioeie aeanoeainoyie, aeiee ia ii/aoeo
60-o ?ie?a a ?iaioao I.Aioi, A.Eieueiaia, I.Oeaoi ? Ae.Noa?iaaeia?a ye
i?ae aieeaii aiaeia?/ii? oaiaoeee c oai??? a?oi, oae ? a ca’yceo c
aeayeeie caaea/aie c oai??? a?oi E?. I?eaeecii a oeae aea /an a ca’yceo
c a?aeiiith oai?aiith Eaaaey-A?eaiaeoa i?i ?ica’yci?noue ne?i/aiii?
a?oie, yea ?iceeaaea?oueny a aeiaooie iiia?ii ia?anoaaieo i?eueiioaioieo
i?aea?oi I.Eaaaeue iinoaaea ieoaiiy i?i oa, /e aoaea ?ica’yciith aeaaa?a
E? L, yea ?iceeaaea?oueny a noio L=A+B nai?o i?eueiioaioieo i?aeaeaaa? A
? A. Oea ieoaiiy, yea noi?ioeueiaaia oaeiae i?ae iiia?ii 5.17 a
a?aeiiiio ca??ieeo ia?ica’ycaieo i?iaeai oai??? a?oi iaei aaeeeee aieea
ia ?icaeoie oe??? oaiaoeee. Aeine?aeaeaiith noi i?eueiioaioieo aeaaa? E?
i?ena’y/aii aaeeeo e?euee?noue ?ia?o, cie?aia ?iaioe I.Aioi,
A.?.Eino?ee?ia, I.Eaaaey, TH.A.Aaooo??ia, A.I.Iao?aa/oea,
I.A.Coniaiiae/a, A.A.Iaitheiaa, A.A.Aaeia, O.E.Oienoiaa oa ?ioeo.
Iaaaoeaiee ?ica’ycie i?iaeaie I.Eaaaey aoa io?eiaiee A.I.Iao?aa/oeii a
1988 ?ioe?, a naia aoea iiaoaeiaaia ia?ica’ycia aeaaa?a E?
oa?aeoa?enoeee p=2, yea ?iceeaaea?oueny a noio aaaeaai? oa
i?eueiioaioii? i?aeaeaaa?. Aeaui i?ci?oa I.A.Coniaiiae/ ? A.A.Iaitheia
aeiaaee ?ica’yci?noue ne?i/aiiiaei??ii? aeaaa?e E? iaae iieai
oa?aeoa?enoeee p>2, yea ?iceeaaea?oueny a noio aeaio nai?o
i?eueiioaioieo i?aeaeaaa? (aeey iiey oa?aeoa?enoeee 0 a?aeiia?aeia
oaa?aeaeaiiy aoei io?eiaii iaaaaaoi ?ai?oa I.Aioi). A ca’yceo c oeei
iniiaiee ?ioa?an i?aaenoaaeythoue iane?i/aiiiaei??i? aeaaa?e E?, ye?
?iceeaaeathoueny a noio aeaio nai?o i?eueiioaioieo aai aeecueeeo aei
i?eueiioaioieo i?aeaeaaa?, ye? aea/aeenue a ?iaioao A.Eieueiaia,
E.I?eeaia, TH.A.Aaooo??ia ? I.Eaaaey, A.I.Iao?aa/oea oa ?ioeo.

Aea/aiith noi aeaio iane?i/aiiiaei??ieo aeaaa? E?, ye? i?eueiioaioi? aai
aeecuee? aei i?eueiioaioieo i?ena’y/aia cia/ia /anoeia ?icae?eo 2
aeena?oaoe?eii? ?iaioe. Cie?aia, aeiaaaeaia ?ica’yci?noue
iane?i/aiiiaei??ii? aeaaa?e E? iaae aeia?eueiei iieai oa?aeoa?enoeee (2,
yea ?iceeaaea?oueny a noio nai?? aaaeaai? oa i?eueiioaioii? i?aeaeaaa?.
Eiiooaoi?ia /eneaiiy, yea ?ic?iaeaia aeey aeiaaaeaiiy oeueiai
?acoeueoaoo iiaea aooe oaeiae ei?eniei ? a oai??? a?oi, aea aiaeia?/iiai
oaa?aeaeaiiy iiee ui iaia?. Iiaeia aeiaanoe, ui ?c ?ica’yciino? aeaaa?e
E? iaae iieai oa?aeoa?enoeee (2 aeaeyaeo L=A+B, aea A aaaeaaa ? A
i?eueiioaioia eeano n aeieeaa?, ui nooi?iue ?ica’yciino? s(L) aeaaa?e L
iaiaaeaiee aeayeith ooieoe??th f(n) a?ae n. Ine?eueee iaa?aeiii, /e
aoaea iaiaaeaiei nooi?iue ?ica’yciino? ne?i/aiii? ?-a?oie G, yea
?iceeaaea?oueny a aeiaooie G=AB aaaeaai? i?aea?oie A ? i?eueiioaioii?
iaaaaeaai? i?aea?oie (iaa?oue eeano 2) i?aea?oie A, oi a?aecia/ai? aeua
?acoeueoaoe c oai??? aeaaa? E? iiaeooue aooe ei?enieie i?e iaeiiio ?c
ianooiieo a?aeiieo i?aeoiae?a:

iaoae G — ne?i/aiia ?-a?oia,

G=d0(G)(d1(G)(…(dn(G)=0

— no?iai oeaio?aeueiee ?yae a?oie G c aeaiaioa?ieie aaaeaaeie oaeoi?aie,
oiaoi

[di(G), dj(G)](di+j(G)

? oaeoi?-a?oia di(G)/ di+1(G) eeaiaioa?ia aaaeaaa (oaeee ?yae ?nio? c
iaeyaeo ia a?aeii? ?acoeueoaoe c oai??? ?-a?oi). Ia i?yi?e noi?

(i=0n-1di(G)/di+1(G)

aaaaeaii a?ia?io iia?aoe?th iiiaeaiiy, aecia/eaoe

[x+ di+1(G), y+dj+1(G)]=[x,y]+di+j+1(G)

aeey aeia?eueieo aeaiaio?a x+di+1(G)(di(G)/di+1(G),
y+dj+1(G)(dj(G)/dj+1(G) ? ?iciianthaeeii iia?aoe?th aeae? ca
e?i?ei?noth. Io?eia?ii, ye a?aeiii, aeaaa?o E? iaae iieai oa?aeoa?enoeee
? ? ?ieiee ?acoeueoaoe aeey aeaaa? E? iiaeooue aooe ia?aianai? ia a?oie.

Ne?i/aii?noue eiiooaioo aeaaa?e E? ? iciaeith aeecueeino? aeaaa?e E? aei
aaaeaai? aeaaa?e ? a ia?aa?ao? 2 ae?oaiai ?icae?eo aea/athoueny noie
aeaio aeaaa? E? c? ne?i/aiiiaei??ieie eiiooaioaie. Cie?aia, aeiaaaeaia
iaeaea ?ica’yci?noue (iane?i/aiiiaei??ii?) aeaaa?e E? iaae aeia?eueiei
iieai, yea ?iceeaaea?oueny a noio aeaio nai?o i?aeaeaaa? c?
ne?i/aiiiaei??ieie eiiooaioaie. A?aecia/eii, ui a oai??? a?oi
caeeoa?oueny a?aee?eoei ieoaiiy i?i iaeaea ?ica’yci?noue aeiaooeo aeaio
a?oi c? ne?i/aiieie eiiooaioaie. *anoeia? aeiaaeee oeueiai ieoaiiy
cono??/aeenue o ??cieo aaoi??a, aea noi?ioeueiaaia aiii yaii a
iiiia?ao?? (ieoaiiy 14a).

Aeecueeeie aei aaaeaaeo aeaaa? E? iiaeia aaaaeaoe oaeiae iaeaea aaaeaa?
aeaaa?e E? (oiaoi oae?, ye? i?noyoue aaaeaaee ?aeaae ne?i/aiii?
eiaei??iino?) ? aea/aiith noi aeaio oaeeo aeaaa? E? i?ena’y/aiee o?ao?e
ia?aa?ao ae?oaiai ?icae?eo aeena?oaoe?eii? ?iaioe. Cie?aia, ye iane?aeie
io?eiaia iaeaea ?ica’yci?noue noie aeaio aeaaa? E?, ne?i/aiiiaei??ieo
iaae nai?ie oeaio?aie (oeae ?acoeueoao, yeee iaeaaeeoue E.I?eeaio, ?
ia?aianaiiyi ia aeaaa?e E? oai?aie I.N.*a?i?eiaa i?i iaeaea
?ica’yci?noue a?oie, yea ?iceeaaea?oueny a aeiaooie aeaio nai?o
i?aea?oi, ne?i/aiieo iaae nai?ie oeaio?aie). Ua iaeiei iane?aeeii
aeuacaaaeaiiai oaa?aeaeaiiy ? iaeaea ?ica’yci?noue noie aaaeaai? oa
iaeaea aaaeaai? aeaaa? E?. A?aoiaoth/e aeeaie? ca’ycee i?ae aeaaa?aie E?
oa e?i?eieie a?oiaie ? c iaeyaeo ia oai?aio ss.I.Nenaea i?i iaeaea
aeainooiaiaao ?ica’yci?noue e?i?eii? a?oie, yea ?iceeaaea?oueny a
aeiaooie aeaio nai?o iaeaea aaaeaaeo i?aea?oi, aoei a oe?eaai ciaeoe
a?aeiia?aeue ia ieoaiiy: /e aoaea noia aaaeaai? oa iaeaea aaaeaai?
aeaaa? E? iaeaea aeainooiaiaai ?ica’yciith, oiaoi, /e aoaea aiia i?noeoe
iaoaaaeaaee ?aeaae ne?i/aiii? eiaei??iino?? O aeiaaeeo aano?aeoieo a?oi
ieoaiiy i?i iaeaea ?ica’yci?noue aeiaooeo aaaeaai? oa iaeaea aaaeaai?
a?oi caeeoa?oueny iiee ui a?aee?eoei , oae naii ye ? a?eueo caaaeueia
ieoaiiy I.N. *a?i?eiaa i?i oa, /e aoaea iaeaea ?ica’yciei aeiaooie aeaio
iaeaea aaaeaaeo a?oi (ieoaiiy 7.55 c? ca??ieea2.

E??i cae/aeieo noi aeaaa? E? a aeena?oaoe?ei?e ?iaio? ?icaeyaeathoueny
ua e oae caai? iio??ei? noie: aeaaa?o E? L aoaeaii iaceaaoe iio??eiith
noiith nai?o i?aeaeaaa? A, B, C, yeui

L=A+B=A+C=B+C.

Oey iinoaiiaea caaea/ iioiaeeoue c oai??? a?oi, aea aeia?a a?aeiia
oai?aia I.Eaaaey i?i i?eueiioaioi?noue ne?i/aiii? a?oie G, yea
?iceeaaea?oueny a iio??eiee aeiaooie G=AB=AC=BC nai?o i?eueiioaioieo
i?aea?oi A, B, C. Aiaeia oeueiai ?acoeueoaoo Eaaaey aaea ia ia? i?noey
aeey aeaaa? E?, a naia ?niothoue ne?i/aiiiaei??i? iai?eueiioaioi?
aeaaa?e E?, ye? ?iceeaaeathoueny a iio??eio noio iaa?oue aaaeaaeo aeaaa?
E?. C aeei?enoaiiyi aeayeeo i?aeoiae?a, aeecueeeo aei oeo ye?
aeei?enoiaoaaeenue ss.I.Nenaeii i?e aeine?aeaeaii? oaeoi?eciaaieo a?oi
c aeiiiiiaith ?aaeeeaeueieo e?eaoeue, io?eiaii iien ne?i/aiiiaei??ieo
aeaaa? E? iaae aeaaa?a?/ii caieiaieie iieyie, ye? ?iceeaaeathoueny a
iio??eio noio aaaeaaeo aeaaa? E?.

I?e aeine?aeaeaii? noie L=A+B i?eueiioaioieo aeaaa? E? A ? B, oae naii
ye ? a oai??? a?oi i?e aea/aii? a?oi, oaeoi?eciaaieo i?eueiioaioieie
i?aea?oiaie, i?e?iaeii aeieea? ieoaiiy i?i ioe?iee nooiaiy ?ica’yciino?
aeaaa?e E? L a caeaaeiino? a?ae eean?a i?eueiioaioiino? aeiaeaie?a A ?
B. Oae naii ye ? aeey a?oi ?nio? ia?ica’ycaia aei oeueiai /ano i?iaeaia
I.Eaaaey2: /e iaiaaeaiee nooi?iue ?ica’yciino? s(L) aeaaa?e E? L noiith
c(A)+c(B) eean?a i?eueiioaioiino? i?aeaeaaa? A ? B. A aeena?oaoe?ei?e
?iaio? iieacaii, ui a oa?aeoa?enoeoe? (2 nooi?iue ?ica’yciino? noie
aaaeaai? aeaaa?e E? oa aeaaa?e E? eeano i?eueiioaioiino? 2 ia ia?aaeuo?
10 (oeae ?acoeueoao aeieeaa? c aeiaaaeaiiy oai?aie i?i ?ica’yci?noue
noie aaaeaai? oa i?eueiioaioii? aeaaa? E? iaae iieai oa?aeoa?enoeee (2).
Oi/a io?eiaia ioe?iea aeineoue a?oaa, aiia ? oe?eaaith oi/a a oiio, ui a
caaaeueiiio aeiaaeeo iaa?aeii? i?ye? ioe?iee aeey aeiaooeo aaaeaai? oa
i?eueiioaioii? eeano 2 a?oi (iaa?oue aeey ne?i/aiieo p-a?oi oaeiai
?iaeo). A?aei?/aii oaeiae, ui aeey aeaaa? E? oa?aeoa?enoeee p=2 acaaae?
ia ?nio? i?yeeo ioe?iie aeey nooiaiy ?ica’yciino? s(L) a caeaaeiino?
a?ae c(A) ? c(B) (iiaoaeiaaii aeaaa?e E? ye caaaiaeii aenieiai nooiaiy
?ica’yciino?, ye? ?iceeaaeathoueny a noio aaaeaai? oa i?eueiioaioii?
eeano 2 i?aeaeaaa?).

Ua iaeei iiaeeeaee oeyo aeey iiaoaeiae eiie?aoieo i?eeeaae?a aeaaa? E? c
oaeoi?ecaoe?yie iiaea aeacaoe a?ioeaaee aeiaooie aeaaa? E? (aeea. ?iaioo
A.E.Oiaeuee?ia). Oea ca’ycaii c oei, ui a?ioeaa? aeiaooee a?oi oni?oii
canoiniaothoueny i?e eiino?othaaii? a?oi c oaeoi?ecaoe?yie ? iiaeia
niiae?aaoenue, ui oey eiino?oeoe?y aoaea ei?enia ? aeey aeaaa? E? c
oaeoi?ecaoe?yie.

Ia?aeaeueii c aeine?aeaeaiiyi aeaaa? E? c oaeoi?ecaoe??th ?icaeaaany
aiaeia?/iee iai?yiie a oai??? anioe?aoeaieo aeaaa? ? e?eaoeue. Iaeieie c
ia?oeo ooo aoee ?acoeueoaoe I.Eaaaey i?i noie i?eueiioaioieo oa noie
ocaaaeueiaii i?eueiioaioieo anioe?aoeaieo aeaaa? oa e?eaoeue, a oaeiae
?acoeueoaoe ?.Oa?noaeia oa E.Niieea i?i PI-e?eueoey, ye?
?iceeaaeathoueny a noio aeaio i?eue-e?eaoeue. Ca inoaii? 10-15 ?ie?a
c’yaeeinue aaaaoi ?ia?o a oeueiio iai?yieo, cie?aia ?iaioe A.Eaea?aaa,
E.?.Aaeaea?a ? A.A.Ieoaeueiaa, TH.A.Aaooo??ia, A.Aeaeaia?oii, I.Eaaaey,
I.Oa??a?i, A.Io/eeianueeiai, A.I.Iao?aa/oea, A.Naeueaa oa ?ioeo, aea
?icaeyaeaeeny noie anioe?aoeaieo aeaaa?, aeecueeeo aei i?eueiioaioieo oa
aei i?eue-aeaaa?.

Iaeia c ieoaiue I.Eaaaey aoei oaeei: /e aoaea i?eue-e?eueoeai
anioe?aoeaia e?eueoea, yea ?iceeaaea?oueny a noio i?eue-i?aee?eueoey oa
eieaeueii i?eueiioaioiiai i?aee?eueoey. Oa?noaei ? Niiee iieacaee, ui
??-e?eueoea eieaeueii i?eueiioaioia, yeui aiii ? noiith aeaio
i?eue-i?aee?eaoeue. A ?iaio?9 aiie iinoaaeee ieoaiiiy i?i oa, /e aoaea
noia aeaio eieaeueii i?eueiioaioieo anioe?aoeaieo e?eaoeue eieaeueii
i?eueiioaioiei e?eueoeai. Iaaaoeaio a?aeieia?aeue ia oea ieoaiiy io?eiaa
A.Eaea?aa, iiaoaeoaaaoe i?eeeaae iai?aa?oie, yea ?iceeaaea?oueny a
ia?aeiaiiy aeaio eieaeueii i?eueiioaioieo iai?aa?oi, aea ia ? eieaeueii
ne?i/aiiith iai?aa?oiith. Iai?aa?oiiaa aeaaa?a oe??? iai?aa?oie iaae
iieai ae?enieo /enae ? aea? i?eeeaae anioe?aoeaii? aeaaa?e, yea
?iceeaaea?oueny a noio aeaio eieaeueii i?eueiioaioieo i?aeaeaaa?, aea ia
? i?eue-aeaaa?ith. Aeaaaioiee i?eeeaae oaeiai ae oeio anioe?aoeaii?
aeaaa?e aeacaa iaaeaaii A.Naeueaa, yeee iiaoaeoaaa iai?aa?oio /anoeiaeo
a?aeia?aaeaiue ae?enii? i?yii? R1 a naaa, yea ?iceeaaea?oueny a
ia’?aeiaiiy aeaio nai?o eieaeueii i?eueiioaioieo i?aeiai?aa?oi, aea
i?noeoue iai?eueiioaioi? aeaiaioe. Aeine?aeaeaiiy a oe?e iaeano?
i?eaa?ioee aei naaa aaeeeo oaaao oaeiae a ca’yceo c oei, ui
A,Io/eeianueeee ? I.Oa??a?i a ?iaio? iieacaee, ui ciaiaieoa i?iaeaia
Eaoa aea?aaeaioia ieoaiith i?i oa, /e aoaea i?eue-e?eueoeai noia
i?eueiioaioiiai e?eueoey oa i?eue-e?eueoey.

Ua iaeei iai?yi a aeine?aeaeaiiyo anioe?aoeaieo aeaaa? c oaeoi?ecaoe?yie
iia’ycaiee c aea/aiiyi noi PI-aeaaa?. Aeae?eueea ?ie?a oiio
TH.A.Aaooo??i oa A.Aeaeaia?oii, a iio?i ? I.Eaaaeue ii/aee aea/aiiy noi
anioe?aoeaieo eiiooaoeaieo e?eaoeue. Ine?eueee oae? noie aeyaeeenue
PI-e?eueoeyie, oi a ca’yceo c oeei E.?.Aaeaea? ? A.A.I?oaeueia a ?iaio?
iinoaaeee ieoaiiy, /e aoaea PI-e?eueoeai anioe?aoeaia e?eueoea, yea
?iceeaaea?oueny a noio aeaio nai?o PI-i?aee?eaoeue? Noiai PI-e?eaoeue
i?ena’y/aii ?iaioe E.?.Aaeaea?a ? A.A.I?oaeaaa, A.I.Iao?aa/oea,
A.Io/eeianueeiai ? I.Eai/eea, ye? c’yaeeenue a inoaii? aeae?eueea ?ie?a.
Aeineoue aoaeoeaiei aia?aoii aeey aeine?aeaeaiiy noi anioe?aoeaieo
aeaaa? ? e?eaoeue aeyaeeanue oai??y ?aaeeeae?a (aeea, iai?eeeaae, eieao
A.A.Aiae?oiae??ae/a ? TH.I.?yaoo?ia). C aeei?enoaiiyi aeayeeo i?aeoiae?a
c oai??? ?aaeeeae?a A.Io/eeianueeee oa I.Eai/ee aeacaee ?yae aaaeeeaeo
aeanoeainoae anioe?aoeaieo e?eaoeue, ye? ?iceeaaeathoueny a noio aeaio
nai?o PI-i?aee?eaoeue c oeie /e ?ioeie aeanoeainoyie. Cie?aia, iaaeaaii
aiie aeiaaee, ui anioe?aoeaia e?eueoea R, yea ?iceeaaea?oueny a noio
R=R1+R2 naiai i?eue-i?aee?eueoey iaiaaeaiiai ?iaeaeno i?eueiioaioiino?
R1 ? i?aee?eueoey R2, yea caaeiaieueiy? aeaye?e iie?iii?aeuei?e
oioiaeiino?, ? ??-e?eueoeai.

Aeine?aeaeaiith noi aeayeeo anioe?aoeaieo e?eaoeue ? aeaaa?
i?ena’y/aiee o?ao?e ?icae?e aeena?oaoe?eii? ?iaioe. Ine?eueee iaeaea
eiiooaoeai? aeaaa?e iaae iieai (oiaoi aeaaa?e, ye? i?noyoue eiiooaoeaiee
?aeaae ne?i/aiii? eiaei??iino?) ? i?e?iaei?i ocaaaeueiaiiyi ye
eiiooaoeaieo aeaaa?, oae ? ne?i/aiiiaei??ieo aeaaa?, oi a
aeena?oaoe?ei?e ?iaio? aeaoaeueii aea/aia aoaeiaa noi oaeeo
anioe?aoeaieo aeaaa?. Iaeei c iniiaieo ?acoeueoao?a ooo — iien
anioe?aoeaii? aeaaa?e iaae aeia?eueiei iieai, yea ?iceeaaea?oueny a noio
aeaio nai?o iaeaea eiiooaoeaieo i?aeaeaaa?. E??i oiai, io?eiaii
ocaaaeueiaiiy a?aeiii? oai?aie I.Eaaaey1 i?i noio i?eueiioaioieo
e?eaoeue: yeui anioe?aoeaia aeaaa?a R iaae aeia?eueiei iieai
?iceeaaea?oueny a noio aeaio nai?o iaeaea i?eueiioaioieo i?aeaeaaa?, oi
aeaaa?a R oaeiae iaeaea i?eueiioaioia.

A?aecia/eii a ca’yceo c iiia?aaei?ie ?acoeueoaoaie, ui noie eieaeueii
i?eueiioaioieo anioe?aoeaieo aeaaa? iiaeooue iaoe aeineoue neeaaeio
aoaeiao. Oea iieacaea I.Ooeoaine?, yea iiaoaeoaaea anioe?aoeaio aeaaa?o,
ui ?iceeaaea?oueny a noio aeaio nai?o eieaeueii i?eueiioaioieo
i?aeaeaaa?, aea i?noeoue a?eueio iaeiiooaoeaio i?aeaeaaa?o. A
aeena?oaoe?ei?e ?iaio? ia ?icaeyaeathoueny anioe?aoeai? aeaaa?e, ui ?
noiith o?ueio nai?o i?aeaeaaa? c oeie /e ?ioeie aeanoeainoyie ?
iia’ycaii oea c oei, ui aoaeiaa ?o iiaea aooe aeoaea neeaaeiith iaa?oue
o aeiaaeeo i?eueiioaioieo aeiaeaie?a: ye iieacaa E.A.Aieooue aoaeue-yea
i?inoa anioe?aoeaia aeaaa?a iiaea aooe aeeaaeaia a noio o?ueio
i?eueiioaioieo anioe?aoeaieo aeaaa?.

E??ai i?eueiioaioi? e?eueoey neeaaeathoue aaaeeeaee i?aeeean eeano
PI-e?eaoeue ? aea/aiiy ?o noi aaaeeeaa oaeiae c oi/ee ci?o oai??? aeaaa?
E? (ia?aoiaey/e aei i?e?aeiaieo e?eaoeue E?, ie aaoiiaoe/ii io?eio?ii
oaa?aeaeaiiy aeey e?eaoeue E?). Aeey noi e??ai i?eueiioaioieo e?eaoeue
iiaeia noi?ioethaaoe ianooiia ieoaiiy (ineaaeaiee aiaeia ieoaiiy Eaaaey
i?i noie aeaaa? E?): /e aoaea e??ai ?ica’yciei anioe?aoeaia e?eueoea,
yea ?iceeaaea?oueny a noio aeaio nai?o e??ai i?eueiioaioieo
i?aee?eaoeue? A aeena?oaoe?ei?e ?iaio? oea ieoaiiy iiceoeaii ?ica’ycaii
o aeiaaeeo, eiee iaeei c aeiaeaie?a eiiooaoeaiee. Io?eiaiee ?acoeueoao
ia aeieeaa? c aiaeia?/ieo aeine?aeaeaiue aeey aeaaa? E?, oi/a a oiio, ui
a?i ni?aaaaeeeaee aeey aeaaa? iaae aeia?eueiei iieai, a ia o?eueee iaae
iieyie oa?aeoa?enoeee (2. C aeiaaaeaiiy aeuacaaaeaiiai ?acoeueoaoo
aeieeaa? oaeiae ioe?iea nooiaiy e??ai? ?ica’yciino? anioe?aoeaiiai
e?eueoey R, yea ?iceeaaea?oueny a noio R=A+B eiiooaoeaiiai i?aee?eueoey
A oa e??ai i?eueiioaioiiai i?aee?eueoey B. Oi/a oey ioe?iea
(eaaae?aoe/iiai oa?aeoa?o) aeineoue aeaeaea a?ae o???, yea i?noeoueny a
a?iioac? I.Eaaaey 2, ?? ?nioaaiiy oe?eaaa c iaeyaeo ia iaeaea iiaio
a?aenooi?noue e?euee?nieo ioe?iie a oe?e oaiaoeoe?.

Ua iaeia caaea/a oaeoi?ecaoe?eiiai oa?aeoa?o, yea iiaea aooe iinoaaeaia
iaeii/anii ye aeey aeaaa? E?, oae ? aeey anioe?aoeaieo aeaaa? ?
iioiaeaeaiiy yei? iia’ycaii a iniiaiiio c oai???th a?oi oi?ioeth?oueny
oaeei /eiii: yeui aeaia aeaaa?a E? L, a ye?e aeey eiaeii? i?aeaeaaa?e A
c aeayei? nenoaie S i?aeaeaaa? c L ?nio? aeiiiaiaiiy, oiaoi oaea
i?aeaeaaa?a B(L (yea caeaaeeoue a?ae i?aeaeaaa?e A), ui L=A+B, A(B=0, oi
ui iiaeia neacaoe i?i aeanoeaino? aeaaa?e L? Aiaeia?/ia caaea/a c oai???
a?oi i?i aea/aiiy a?oi c nenoaiaie aeiiiaithaaieo i?aea?oi a?aeiineoueny
aei iai?yio, iia’ycaiiio c ?i’yi N.I.*a?i?eiaa. Ne?i/aii? a?oie, a yeeo
aeiiiaithaai? an? i?aea?oie (oe?eeii oaeoi?eciaai? a?oie) aea/aeenue ua
a 30-o ?ieao O.Oieeii, a i?ci?oa aeia?euei?, ye ne?i/aii?, oae ?
iane?i/aii?, oe?eeii oaeoi?eciaai? a?oie aoee iienai? I.A.*a?i?eiaith. A
ca’yceo c oeei oe?eaaei ? ieoaiiy i?i aoaeiao aeaaa? E?, a yeeo
aeiiiaithaai? an? i?aeaeaaa?e. sse iieacaii a aeena?oaoe?ei?e ?iaio?,
aeaaa?e E? oaeiai ?iaeo ? ye?ac aeaaa?aie E? c ioeueiaei ia?aoeiii an?o
i?aeaeaaa? eiaei??iino? 1. Aeaaa?e E? ioeueiai? oa?aeoa?enoeee c oaeith
oiiaith aea/aeenue aaaaoueia aaoi?aie (AE.O?on, E.Aioiai, Ae.Iiaoioea oa
?io?) a ca’yceo c aeayeeie caaea/aie ?c oai??? a?oi E? oa iai?aaeaaa?
E?. Caaea/a iieno ne?i/aiiiaei??ieo ae?enieo aeaaa? E? c ioeueiaei
ia?aoeiii an?o i?aeaeaaa? eiaei??iino? 1 aea?aaeaioia iieno ae??
ne?i/aiiiaei??ii? a?oie E? ia iaeiiaei??ieo aeaaeeeo iiiaiaeaeao, yea
aoei io?eiaia N.E? a ua 70-o ?ieao ieioeiai noie?ooy. Oiio oe?eaaei ?
ieoaiiy i?i aoaeiao aeaaa? E? aeacaiiai aeaeo iaae iieyie iiceoeaii?
oa?aeoa?enoeee.

A aeena?oaoe?ei?e ?iaio? iienai? ne?i/aiiiaei??i? aeaaa?e E? iaae
aeineiiaeeie iieyie iiceoeaii? oa?aeoa?enoeee ? iaae aeia?eueiei iieai
oa?aeoa?enoeee p=2, a yeeo aeiiiaithaai? an? iaeiiaei??i? i?aeaeaaa?e ?
oei naiei aeyniaii aoaeiao aeaaa? E? c ioeueiaei ia?aoeiii an?o
i?aeaeaaa? eiaei??iino? 1 iaae aeacaieie iieyie. I?e aeayeeo iaiaaeaiiyo
aeaii oaeiae iien iane?i/aiiiaei??ieo eieaeueii ne?i/aiieo aeaaa? E? c
aeiiiaithaaieie iaeiiaei??ieie i?aeaeaaa?aie. Oe?eaai a?aecia/eoe, ui ye
? a aeiaaeeo noi i?eueiioaioieo aeaaa? E?, ooo oa?aeoa?enoeea iniiaiiai
iiey oaeiae nooo?ai aieeaa? ia no?oeoo?o aeaaa? E?, ye? aea/athoueny
(iaae iieyie oa?aeoa?enoeee p=2 aeaaa?e E? c aeiiiaithaaieie
i?aeaeaaa?aie ?ica’yci?, ia i?ioeaaao oa?aeoa?enoeoe? (2). Io?eiai?
?acoeueoaoe iiaeooue aooe oaeiae ei?enieie i?e iien? ae??
iane?i/aiiiaei??ieo a?oi E? ia aeaaeeeo iiiaiaeaeao. sse iieacaii a
aeena?oaoe?ei?e ?iaio? oiiaa aeiiiaithaaiino? iaeiiaei??ieo i?aeaeaaa?
ne?i/aiiiaei??ii? aeaaa?e aea?aaeaioia oiia? aeiiiaithaaiino? an?o ??
i?aeaeaaa?. Aiaeia?/ia oaa?aeaeaiiy aaea ia ia? i?noey aeey
iane?i/aiiiaei??ieo aeaaa? E? ? oiio iio??aii aea/aoe aaea aeaa ??cieo
eeane aeaaa? E? (eean aeaaa? E?, a yeeo aeiiiaithaai? an? i?aeaeaaa?e —
n-aeaaa? E? ? eean aeaaa? E? c aeiiiaithaaieie iaeiiaei??ieie
i?aeaeaaa?aie — n1-aeaaa? E?, ye aiie aeey c?o/iino? iaceaathoueny a
aeena?oaoe?ei?e ?iaio?.).

Iane?i/aiiiaei??i? aeaaa?e E?, ye? aea/athoueny a aeena?oaoe?ei?e
?iaio?, aaaaeathoueny ne??cue eieaeueii ne?i/aiieie (oiaoi eieaeueii
ne?i/aiiiaei??ieie) ? oiio a?aeiia?aeue ia ieoaiiy: /e aoaea aeaaa?a E?
c aeiiiaithaaieie i?aeaeaaa?aie eieaeueii ne?i/aiiith? — ciyea a
iaiaaeaiiy a oi?ioethaaiiyo io?eiaieo ?acoeueoao?a. A?aeiia?aeue ia
aiaeia?/ia ieoaiiy aeey aeaaa? E? c aeiiiaithaaieie iaeiiaei??ieie
ieoaiiyie iaaaoeaia (aaoi? aaey/iee i?io. E.Iaaao ca ianooiiee
i?eeeaae): iaoae L — aeaaa?a E? an?o aeaaeeeo aaeoi?ieo iie?a ia
ae?eni?e i?yi?e, Lx={v(L (v(x)=0} — i?aeaeaaa?a oeo iie?a, ye? iathoue
ioeueiaa cia/aiiy a oi/oe? x(R. Iaaaaeei ia?aeiiaoenue, ui
oaeoi?-i?ino?? L/Lx iaeiiaei??iee ? ia?aoei an?o i?aeaeaaa? Lx aeey x(R
ioeueiaee. Oiio, ye iieacaii a aeena?oaoe?ei?e ?iaio?, a aeaaa?? L
aeiiiaithaai? an? iaeiiaei??i? i?aeaeaaa?e, oi/a aeaaa?a L ? ia ?
eieaeueii ne?i/aiiiaei??iith.

Aeey aeaaa?e E? L c ioeueiaei ia?aoeiii an?o i?aeaeaaa? eiaei??iino? 1
i?aeaeaaa?a O?aoo?i? ((L) (oiaoi ia?aoei an?o iaeneiaeueieo i?aeaeaaa?
?c L), i/aaeaeii, aei??aith? ioeth. Ne?i/aiiiaei??i? aeaaa?e E? c ((L)=0
(ia?aaaaeii ioeueiai? oa?aeoa?enoeee) aeine?aeaeoaaeenue a aaaaoueio
?iaioao (Aa?ina, Oaoa?na, Oo?ooe?iaa?a oa ?ioeo) ? oiio aeayeo
?ioi?iaoe?th i?i aoaeiao aeaaa? E? oa?aeoa?enoeee 0 c aeiiiaithaaieie
iaeiiaei??ieie i?aeaeaaa?aie iiaeia io?eiaoe ?c ?o ?ia?o.

Aea/aiiy aeaaa? E? c nenoaiaie aeiiiaithaaieo i?aeaeaaa?, ye
a?aecia/aeiny aeua, o?nii iia’ycaia c aiaeia?/iith, aeineoue ?icaeiaiith
oaiaoeeith a oai??? a?oi, ye a ?aeaeiiio a?aeiioaii?, oae ?
aeei?enoaiiyi aeayeeo i?aeoiae?a ? iaoiae?a. Cie?aia, oea i?aeoiaee, ye?
aeei?enoiaothoueny i?e aea/aii? a?oi, a yeeo aeiiiaithaai? an? oeeee?/i?
i?aea?oie, an? iaoeeee?/i? i?aea?oie ? o.ae. O?ao?e ia?aa?ao /aoaa?oiai
?icae?eo aeena?oaoe?eii? ?iaioe i?ena’y/aii naia aea/aiith a?oi, ye
ne?i/aiieo, oae ? iane?i/aiieo c aeayeeie oiiaaie ia nenoaio
aeiiiaithaaieo i?aea?oi. Io?eiaii iien iane?i/aiieo a?oi c
aeiiiaithaaieie iaoeeee?/ieie iai?eia?ieie i?aea?oiaie ?
ioa?aeoa?eciaai? (i?e aeayeeo iaiaaeaiiyo) ne?i/aii? a?oie c
aeiiiaithaaieie iaaenee?anueeeie i?aea?oiaie (oiaoi oaeeie i?aea?oiaie,
ye? i?nothoue oi/a a iaeio iaiaeeie/io neeianueeo i?aea?oio an???
a?oie).

E??i oiia aeiiiaithaaiino? ia nenoaio i?aeaeaaa? aeaaa?e (aai nenoaio
i?aea?oi a?oie) iiaeii iaeeaaeaoe iaiaaeaiiy ? ?ioiai oa?aeoa?o,
ca’ycaiiai, iai?eeeaae, c ?ica’yci?noth aai c i?eueiioaioi?noth
aeaiaio?a oe??? nenoaie. Oa, ui oeae i?aeo?ae ? ie?aeiei ia aeeeeea?
noii?ao, ine?eueee oae? aaaeeea? iiiyooy, ye eieaeueia ?ica’yci?noue,
eieaeueia i?eueiioaioi?noue, iaeaea ?ica’yci?noue ? o.ae. aeieeathoue
naia ia oeueiio oeyoo. Ia i?ioeaaao oai??? a?oi, a oai??? aeaaa? E?
oaiaoeea, yea ca’ycaia c ocaaaeueiaiith ?ica’yci?noth oa
i?eueiioaioi?noth ?icaeiaia aeineoue neaai ? ine?eueee aeaaa?e E? oaeiai
aeaeo aeieeathoue i?e?iaei?i /eiii i?e aea/aii? noi i?eueiioaioieo aai
aeecueeeo aei i?eueiioaioieo aeaaa? E?, iaeei c ia?aa?ao?a aeena?oaoe??
i?enay/aii aea/aiith iaeaea ?ica’ycieo aeaaa? E?. Iaeaea ?ica’yci?
aeaaa?e E?, oiaoi aeaaa?e E?, ye? i?noyoue ?ica’yciee ?aeaae ne?i/aiii?
eiaei??iino?, ? i?e?iaei?i ocaaaeueiaiiyi ?ica’ycieo aeaaa? E? ? aeaaa?
E? ne?i/aiii? aei??iino?. Oiio i?e?iaeiuei i/?eoaaoe, ui aeey ieo
ni?aaaaeeea? aiaeiae ?acoeueoao?a i?i ne?i/aiiiaei??i? oa ?ica’yci?
aeaaa?e E?.

Anoaiiaeaii aeaye? aaaeeea? aeanoeaino? iaeaea ?ica’ycieo aeaaa? E?, ye?
aiaeia?/i? aeanoeainoyi iaeaea ?ica’ycieo a?oi (aeaaa?a E? c ?ica’yciith
i?aeaeaaa?ith ne?i/aiii? eiaei??iino? iaeaea ?ica’ycia, ?icoe?aiiy
iaeaea iaeaea ?ica’ycii? aeaaa?e E? ca aeiiiiiaith iaeaea ?ica’ycii?
aeaaa?e E? ? iaeaea ?ica’yciith ? o.ae.) I?e oeueiio aeacai? i?eeeaaee,
eiee oaei? aiaeia?? aaea iaia?: iai?eeeaae, aaaeaaa i?aeaeaaa?a
ne?i/aii? eiaei??iino? a iane?i/aiiiaei??i?e aeaaa?? E? iiaea ia i?noeoe
iaioeaaeo ?aeaae?a an??? aeaaa?e. Aeia?a a?aeiii, ui ?ica’yciee ?aaeeeae
ne?i/aiiiaei??ii? aeaaa?e E? ioeueiai? oa?aeoa?enoeee ? oa?aeoa?enoe/iei
?aeaaeii oe??? aeaaa?e ? ui oea aaea iaa??ii aeey aeaaa? E? iiceoeaii?
oa?aeoa?enoeee. Oiio oe?eaaith ? oa ianoaaeia, ui oey ??cieoey aaea
cieea? i?e ia?aoiae? aei iaeaea ?ica’ycieo aeaaa? E?. Caoaaaeeii, ui
aeiaaaeaiiy a?eueoino? ?acoeueoao?a i?i iaeaea ?ica’yci? aeaaa?e E?
a?oioothoueny ia i?aeoiaeao, ye? i?eioeeiiai a?ae??ciythoueny a?ae
i?aeoiae?a i?e aea/aii? iaeaea ?ica’ycieo a?oi.

E??i aeaaa? E? c iaiaaeaiiyie ia nenoaie ?ica’ycieo i?aeaeaaa?
?icaeyaeathoueny oaeiae ia??iaee/i? eieaeueii ?ica’yci? a?oie c
iaiaaeaiiyie ia neeianuee? (-i?aea?oie. Aeey ?o aea/aiiy anoaiiaeaii
aeaye? caeaaeiino? a ne?i/aiieo (-?ica’ycieo a?oiao (( — aeayea
ne?i/aiia iiiaeeia i?inoeo /enae) i?ae aeiaaeeiaie (’-(-?yae?a ?
ia?aiao?aie oieeianueeeo (-i?aea?oi. Cie?aia aeiaaaeaii, ui a
ia??iaee/i?e eieaeueii ?ica’yci?e a?oi? G an? neeianuee? i?aea?oie
iathoue ne?i/aii? aeniiiaioe oiae? ? o?eueee oiae?, eiee aiia ia?
ne?i/aiiee ?yae oa?aeoa?enoe/ieo i?aea?oi, eiaeiee oaeoi? yeiai ? aai
?-a?oiith aeey aeayeiai ?(( aai (’-a?oiith. Ca?aene, cie?aia, aeieeaa?,
ui eieaeueii ?ica’ycia a?oia ne?i/aiii? aeniiiaioe ia? ne?i/aiiee ?yae
oa?aeoa?enoe/ieo i?aea?oi c i?eia?ieie oaeoi?aie. Aeacai? oaeiae aeaye?
aeanoeaino? oaeeo a?oi, ye? iia’ycai? c iayai?noth a ieo ?yae?a c
i?eueiioaioieie aai eieaeueii i?eueiioaioieie (-oaeoi?aie.

Iaoa ?iaioe. Iaoith ?iaioe ? aeine?aeaeaiiy aeaaa? E?, ye
ne?i/aiiiaei??ieo oae ? iane?i/aiiiaei??ieo, ye? ?iceeaaeathoueny a noio
aeaio nai?o i?eueiioaioieo aai aeecueeeo aei i?eueiioaioieo i?aeaeaaa?,
aea/aiiy anioe?aoeaieo aeaaa? ? e?eaoeue, ye? ?iceeaaeathoueny a noio
aeaio i?aeaeaaa? (i?aee?eaoeue) c oiiaaie, aeecueeeie aei eiiooaoeaiino?
aai i?eueiioaioiino?, io?eiaiiy ioe?iie aeey nooiaiy ?ica’yciino?
aeaaa?e L=A+B a caeaaeiino? a?ae eean?a i?eueiioaioiino? aeiaeaie?a A ?
A, aeine?aeaeaiiy aeaaa? E?, c aeiiiaithaaieie i?aeaeaaa?aie a oaeiae c
aeiiiaithaaieie iaeiiaei??ieie i?aeaeaaa?aie, anoaiiaeaiiy aeanoeainoae
iaeaea ?ica’ycieo aeaaa? E? (oiaoi aeaaa? E?, ye? i?noyoue ?ica’yciee
?aeaae ne?i/aiii? eiaei??iino?), a oaeiae aeayeeo eean?a a?oi,
ia?aaaaeii iane?i/aiieo c aeayeeie nenoaiaie aeiiiaithaaieo i?aea?oi ?
aeanoeainoyie neeianueeeo (-i?aea?oi.

Iaoeiaa iiaecia. A aeena?oaoe?ei?e ?iaio? aaoi?ii io?eiai? iia?
oai?aoe/i? ?acoeueoaoe, cie?aia:

aeaii iaaaoeaiee ?ica’ycie i?iaeaie I.Eaaaey i?i oa, /e aoaea
?ica’yciith aeaaa?a E?, yea ?iceeaaea?oueny a noio aeaio nai?o
i?eueiioaioieo i?aeaeaaa?;

aeiaaaeaii ?ica’yci?noue (iane?i/aiiiaei??ii?) aeaaa?e E? iaae iieai
oa?aeoa?enoeee (2, yea ?iceeaaea?oueny a noio aaaeaai? oa i?eueiioaioii?
i?aeaeaaa?,

anoaiiaeaii iaeaea ?ica’yci?noue aeaaa?e E? iaae aeia?eueiei iieai, yea
?iceeaaea?oueny a noio aeaio nai?o i?aeaeaaa? c? ne?i/aiiiaei??ieie
eiiooaioaie;

aeiaaaeaia iaeaea ?ica’yci?noue noie iaeaea aaaeaai? aeaaa?e E? oa
aeaaa?e E?, yea ne?i/aiiiaei??ia iaae nai?i oeaio?ii, ? ye iane?aeie
io?eiaii iaeaea ?ica’yci?noue noie aaaeaai? oa iaeaea aaaeaai? aeaaa?
E?.

anoaiiaeaii iniiai? aeanoeaino? iaeaea ?ica’ycieo aeaaa? E?: iieacaii,
ui oaea aeaaa?a E? i?noeoue oa?aeoa?enoe/iee ?ica’yciee ?aeaae
ne?i/aiii? eiaei??iino?, aeiaaaeaii, ui aeaaa?a E? c ?ica’yciith
i?aeaeaaa?ith ne?i/aiii? eiaei??iino? iaeaea ?ica’ycia;

aeine?aeaeaii aoaeiao anioe?aoeaieo aeaaa? iaae aeia?eueiei iieai, ye?
?iceeaaeathoueny a noio aeaio nai?o iaeaea eiiooaoeaieo i?aeaeaaa?,
aeiaaaeaii e??ao ?ica’yci?noue noie eiiooaoeaiiai oa e??ai
i?eueiioaioiiai anioe?aoeaieo e?eaoeue;

aeaii iien ne?i/aiiiaei??ieo aeaaa? E? iaae aeineiiaeei iieai, a ye?e
aeiiiaithaaia eiaeia i?aeaeaaa?a oa ioa?aeoa?eciaai? aeaye? eeane
iane?i/aiiiaei??ieo aeaaa? E? c aeiiiaithaaieie iaeiiaei??ieie
i?aeaeaaa?aie;

aeine?aeaeaii iane?i/aii? a?oie c aeiiiaithaaieie iaoeeee?/ieie
iai?eia?ieie i?aea?oiaie ? (i?e aeayeeo iaiaaeaiiyo) ne?i/aii? a?oie c
aeiiiaithaaieie iaaenee?anueeeie i?aea?oiaie oa ioa?aeoa?eciaai?
ia??iaee/i? eieaeueii ?ica’yci? a?oie c ne?i/aiiiaeniiiaioieie
nee?anueeeie (-i?aea?oiaie;

An? oe? ?acoeueoaoe io?eiaii aia?oa.

Oai?aoe/ia oa i?aeoe/ia oe?ii?noue aeena?oaoe??. ?iaioa ia? oai?aoe/iee
oa?aeoa?. ?acoeueoaoe iiaeooue aooe aeei?enoai? a iiaeaeueoeo
aeine?aeaeaiiyo c oai??? iane?i/aiiiaei??ieo aeaaa? E? c
oaeoi?ecaoe?yie, i?e aea/aii? noi anioe?aoeaieo aeaaa? ? e?eaoeue.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. ?acoeueoaoe aeena?oaoe?eii?
?iaioe a ??ciee /an aeiiia?aeaeenue aeiiia?aeaeenue:

ia 10-iio Ananithciiio neiiic?oi? c oai??? a?oi (Aiiaeue 1986?),

ia XIX Ananithci?e aeaaa?a?/i?e eiioa?aioe?? (i.Euea?a, 1987?.),

ia I?aeia?iaei?e aeaaa?a?/i?e eiioa?aioe?? iai’yo? A.?.Iaeueoeaaa
(Iiainea??nuee, 1989?.),

ia I?aeia?iaei?e aeaaa?a?/i?e eiioa?aioe?? iai’yo? A.?.Oe?oiaa (Aa?iaoe,
1991 ?.),

ia I?aeia?iaeieo iaoaiaoe/ieo eiioa?aioe?yo iai’yo? I.I.E?aa/oea (Ee?a,
1992, 1996),

ia Anaoe?a?inuee?e iaoaiaoe/i?e eiioa?aioe?? iaiyo? iai’yo?
I.Eacei??nueeiai (Euea?a, 1995),

ia 5-e I?aeia?iaei?e eiioa?aioe?? “A?oie ? a?oiia? e?eueoey” (Euea?a,
1996),

ia I?aeia?iaei?e aeaaa?a?/i?e eiioa?aioe?? iai’yo? Ae.E.Oaaeaeaaaa
(N.-Iaoa?ao?a, 1997),

ia I?aeia?iaei?e aeaaa?a?/i?e eiioa?aioe?? iai’yo? E.I.Aeone?ia
(Neiayinuee, 1997).

E??i oiai, ?acoeueoaoe aeena?oaoe?eii? ?iaioe aeiiia?aeaeenue ia
nai?ia?? eaoaae?e aeaaa?e Iineianueeiai aea?aeaaiiai oi?aa?neoaoo (1987
?.), ia nai?ia?? c oai??? a?oi Iiainea??nueeiai oi?aa?neoaoo (1989 ?.),
ia aeaaa?a?/iiio nai?ia?? ?inoeoooo iaoaiaoeee oi?aa?neoaoo i.O?aeao?a
(O?I, 1996), ia nai?ia?? c aeaaa?e oi?aa?neoaoo i.Eieueia? (O?aioe?y,
1996), ia aeaaa?a?/iiio nai?ia?? Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana
Oaa/aiea (1998 ?.), ia aeaaa?a?/iiio nai?ia?? ?inoeoooo iaoaiaoeee IAI
Oe?a?ie (1999 ?.).

Ioae?eaoe??. Iniiai? ?acoeueoaoe aeena?oaoe?? iioae?eiaai? a 15
iaoeiaeo noaooyo, a oaeiae a 12 oacao aeiiia?aeae iaoeiaeo eiioa?aioe?e
(oea ioae?eaoe?? [1] — [15] oa a?aeiia?aeii [16]—[27] ?c nieneo
?ia?o, yeee iaaaaeaii a e?ioe? aaoi?aoa?aoo).

-?ac?aoeiuo a?oii” aaoi?o iniaenoi iaeaaeeoue /anoeia 2 oai?aie 2, yea
iia’ycaia c aeiaaeeii aaaeaaeo 2-i?aea?oi a?oie. Aea? ?io? ni?euei?
?iaioe ? oacaie eiioa?aioe?e, aea aoee aiiiniaai? ?acoeueoaoe aeaio
aeuacaaaeaieo ni?eueieo ?ia?o.

No?oeoo?a oa ia’?i aeena?oaoe??. ?iaioa neeaaea?oueny c anooio,
?icae?eo “Iaeyae e?oa?aoo?e. Iiia?aaei? aeai?”, ?icae?eo “Noie
i?eueiioaioieo /e aeecueeeo aei i?eueiioaioieo aeaaa? E?” (i?noeoue 4
i?ae?icae?ee), ?icae?eo “Noie anioe?aoeaieo aeaaa?” (i?noeoue 2
i?ae?icae?ee), ?icae?eo “Aeaaa?e E? ? a?oie c nenoaiaie aeiiiaithaaieo
i?aeaeaaa? ? i?aea?oi” (i?noeoue 3 i?ae?icae?ee), ?icae?eo “Aeaaa?e E? ?
a?oie c aeayeeie nenoaiaie ?ica’ycieo i?aeaeaaa? ? i?aea?oi” (i?noeoue 2
i?ae?icae?ee), caaaeueieo aeniiae?a, nienea e?oa?aoo?e, ui neeaaea?oueny
c 145 iaeiaioaaiue, caaaeueiee ianya ?iaioe 274 noi??iee

CI?NO ?IAIOE

O anooi? aea?oueny iaeyae iai?yie?a a oai??? aeaaa? E? ? anioe?aoeaieo
aeaaa?, iia’ycaieo c iiiyooyi oaeoi?ecaoe??. A ia?oiio ?icae?e? aeaii
iaeyae e?oa?aoo?e ca oaiith aeena?oaoe?? oa aeeeaaeai? iaoiaee
aeine?aeaeaiiy aeaaa? E? oa anioe?aoeaieo aeaaa? c oaeoi?ecaoe?yie..
Iaaiaeyoueny oaeiae iniiai? ?acoeueoaoe, ye? aeei?enoiaothoueny a
aeena?oaoe?ei?e ?iaio?. Aeathoueny icia/aiiy aeayeeo iaoaiaoe/ieo
ia’?eo?a ? iiiyoue, ye? aeei?enoiaothoueny a ?iaio?, aea ia ?
eaaeiaeinooiieie a e?oa?aoo??.

A ae?oaiio ?icae?e? aea/athoueny noie aeaaa? E?, ye? aai i?eueiioaioi?
aai aeecuee? aei i?eueiioaioieo. A ia?oiio ia?aa?ao? aea?oueny ?ica’ycie
(iaaaoeaiee) a?aeiii? i?iaeaie I.Eaaaey i?i ?ica’yci?noue aeaaa?e E?,
yea ?iceeaaea?oueny a noio aeaio nai?o i?eueiioaioieo i?aeaeaaa?:

Oai?aia 2.1.1. Iaae aeia?eueiei iieai E oa?aeoa?enoeee ?=2 ?nio?
ia?ica’ycia ne?i/aiiiaei??ia aeaaa?a E? L, yea ?iceeaaea?oueny a noio
L=A+B aaaeaai? i?aeaeaaa?e A ? i?eueiioaioii? (eeano 2) i?aeaeaaa?e A.

Oeae ?acoeueoao ? iniiaiei a ia?aa?ao? 1 ae?oaiai ?icae?eo. Ooo oaeiae
?icaeiaii aeayea eiiooaoi?ia /eneaiiy, yea oe?iei aeei?enoiao?oueny a
iiaeaeueoeo ?icae?eao aeena?oaoe??. I?eeeaaeii oaa?aeaeaiiy i?i aeiaooee
aeaiaio?a a aeaaa?ao E?, ui ?iceeaaeathoueny a noio aeaio nai?o
i?aeaeaaa? iiaea neoaoaaoe ianooiia

Eaia 2.1.3. Iaoae L- aeaaa?a E? iaae anioe?aoeaii-eiiooaoeaiei e?eueoeai
c iaeeieoeath, yea ?iceeaaea?oueny a noio L=A+B aaaeaai? i?aeaeaaa?e A ?
aeayei? i?aeaeaaa?e A. Oiae? aeey aoaeue-yeeo iaoo?aeueieo /enae i, j
eeiiothoueny ni?aa?aeiioaiiy:

[[A, Bi],[A, Bj]]( [A, Bi+1]+ [A, Bj+1]+ [A, Bi+j,L]+B2

[A, Bi+j,L]( [A, Bi]+ B2

C oeeo ni?aa?aeiioaiue eaaei io?eiaoe oaeiae ?ica’yci?noue noie aeaio
aaaeaaeo aeaaa? E? (aiaeia oai?aie I.?oi c oai??? a?oi), yea aoea
anoaiiaeaia A.Eieueiaiii.

C aeei?enoaiiyi aeuacaaaeaiiai eiiooaoi?iiai /eneaiiy anoaiiaeaii oaeiae
oaea oaa?aeaeaiiy i?i noio aaaeaai? oa i?eueiioaioii? aeaaa? E?:

Oai?aia 2.1.4. Iaoae L — iaioeaaa aeaaa?a E? iaae anioe?aoeaii-
eiiooaoeaiei e?eueoeai R c iaeeieoeath, yea ?iceeaaea?oueny a noio
L=A+B aaaeaai? i?aeaeaaa?e A ? i?eueiioaioii? i?aeaeaaa?e A. Oiae?
aeaaa?a L ia ni?aiaaea? c? nai?i eiiooaioii.

Oey oai?aia, cie?aia, iieaco?, /iio eiio?i?eeeaae aei a?iioace I.Eaaaey
c oai?aie 2.1.1 ? iai?inoith aeaaa?ith E?. C oai?aie 2.1.1. iiaeia,
aeei?enoiaoth/e eiino?oeoe?th oaici?iiai aeiaooeo aeaaa?e E? oa
anioe?aoeaii eiiooaoeaii? aeaaa?e iaae oei aea iieai, io?eiaoe oaeee
iane?aeie, yeee ia? naiino?eia cia/aiiy ? ua ?ac i?aee?aneth? niaoeeo?eo
oa?aeoa?enoeee ?=2 i?e aea/aii? noi aeaaa? E?.

Oaa?aeaeaiiy 2.1.6. Iaae aeia?eueiei iieai E oa?aeoa?enoeee ?=2 ?nio?
ne?i/aiiiaei??ia aeaaa?a E? L aeia?eueiiai iaia?aae caaeaiiai nooiaiy
?ica’yciino? n, yea ?iceeaaea?oueny a noio L=A+B aaaeaai? i?aeaeaaa?e A
oa i?eueiioaioii? eeano 2 i?aeaeaaa?e A.

Ua iaeiei aaaeeeaei ?acoeueoaoii ia?aa?aoo 1 ?icae?eo 2 ? ianooiia
oai?aia:

Oai?aia 2.1.10. Iaoae L — (iane?i/aiiiaei??ia) aeaaa?a E? iaae iieai E
oa?aeoa?enoeee (2, yea ?iceeaaea?oueny a noio L=A+B aaaeaai?
i?aeaeaaa?e A oa i?eueiioaioii? i?aeaeaaa?e A. Oiae? aeaaa?a L
?ica’ycia.

Aeey aeiaaaeaiiy oe??? oai?aie a aeaaa?? E? L=A+B aoaeo?oueny
i?aeaeaaa?a aeaeo L1=N+B, aea E-i?aei?ino?? N ia? aeanoeaino? [N, N](B2
, [N, B](N, N(B=0, oiaoi L1 ? Z2-a?aaeoeiaaiith aeaaa?ith E?.
Aeei?enoiaoth/e aeaye? ni?aa?aeiioaiiy neiao??? a oe?e i?aeaeaaa??
ioe?ith?oueny nooi?iue ?ica’yciino? ?? ?aeaaeo c ?ica’yciith
oaeoi?-aeaaa?ith. Ia aeaeue, oeae i?aeo?ae ia aea? caaaeueieo ioe?iie
aeey nooiaiy ?ica’yciino? aeaaa?e E? L a caeaaeiino? a?ae eeano
i?eueiioaioiino? i?aeaeaaa?e A. Aea ana ae oaee, c aeei?enoaiiyi
caaaeueii? noaie aeiaaaeaiiy oai?aie 2.1.10, io?eiaii oaea oaa?aeaeaiiy
(oe?eaaith ooo ? ae?oaa /anoeia oaa?aeaeaiiy):

Oaa?aeaeaiiy 2.1.12. sseui aeaaa?a E? L iaae iieai K oa?aeoa?enoeee (2
?iceeaaea?oueny a noio L=A+B aaaeaai? i?aeaeaaa?e A oa i?eueiioaioii?
eeano n i?aeaeaaa?e A, oi L ?ica’ycia ? nooi?iue ?? ?ica’yciino?
s(L)(f(n) aeey aeayei? ooieoe?? f(n). sseui i?aeaeaaa?a A i?eueiioaioia
eeano 2, oi s(L)(10.

Iaiaaeaiiy ia oa?aeoa?enoeeo iiey a oeueiio iane?aeeo nooo?aa ? ia
o?eueee oiio, ui a oa?aeoa?enoeoe? ?=2 ?niothoue ia?ica’yci? aeaaa?e E?,
ye? ?iceeaaeathoueny a noio aaaeaai? oa i?eueiioaioii? i?aeaeaaa?, aea
oaeiae c iaeyaeo ia aeuaiaaaaeaia oaa?aeaeaiiy 2.1.6.

A ia?aa?ao? 2.2 aea/athoueny aeaaa?e E? (iane?i/aiiiaei??i?), ye?
?iceeaaeathoueny a noio aeaio i?aeaeaaa? c? ne?i/aiiiaei??ieie
eiiooaioaie. Aeaaa?e E? L c dim[L, L]<( aeecuee? aei aaaeaaeo, ine?eueee aaee/eia eiiooaioo [L, L] ? i??ith iaaaaeaaino? aeaaa?e E? L. Oiio ieoaiiy i?i no?oeoo?o aeaaa?e E? L, yea ? noiith aeaio nai?o i?aeaeaaa? A ? A c dim[A, A]+dim[B, B]<( ? i?e?iaei?i. Iniiaiei ?acoeueoaoii oeueiai ia?aa?aoo ? Oai?aia 2.2.20. Iaoae L - aeaaa?a E? iaae aeia?eueiei iieai, yea ?iceeaaea?oueny a noio L=A+B nai?o i?aeaeaaa? A ? A c? ne?i/aiiiaei??ieie eiiooaioaie. Oiae? aeaaa?a L iaeaea ?ica’ycia. I?e aeiaaaeaii? oe??? oai?aie oe?iei aeei?enoiaoaaeeny aeanoeaino? FC-aeaaa? E? (aeaaa? E?, a yeeo eiaeai aeaiaio ia? oeaio?ae?caoi? ne?i/aiii? eiaei??iino?), ine?eueee a aeaaa?? E? c? ne?i/aiiiaei??iei eiiooaioii, ye iaaaaeei ia?aeiiaoenue, oeaio?ae?caoi? eiaeiiai aeaiaioo ia? ne?i/aiio eiaei??i?noue. Aeaye? c oeeo aeanoeainoae io?eiai? a aeena?oaoe?ei?e ?iaio?, aeaye? aoee anoaiiaeai? ?ai?oa ?ioeie aaoi?aie aeey oae caaieo ?aeaaeueii ne?i/aiieo aeaaa? E? (a yeeo eiaeai aeaiaio i?noeoueny a ne?i/aiiiaei??iiio ?aeaae?), ye?, iani?aaae?, ? FC-aeaaa?aie E?. Ianooiia eaia aeaco? aeaye? aaaeeea? aeanoeaino? aeaaa? E? c? ne?i/aiiiaei??ieie eiiooaioaie, ye? /anoi aeei?enoiaothoueny i?e aea/aii? noi aeaaa? E?. Eaia 2.2.7. Iaoae L - aeaaa?a E? c? ne?i/aiiiaei??iei eiiooaioii [L, L]. Oiae? ni?aaaaeeea? oaa?aeaeaiiy: (a) L ? FC-aeaaa?ith E?; (a) yeui [L, L](Z(L)=0, oi L=A(B, aea A - aeayeee aaaeaaee ?aeaae ? A - aeayeee ne?i/aiiiaei??iee ?aeaae aeaaa?e L; (a) aeaaa?a L i?noeoue oa?aeoa?enoe/iee i?eueiioaioiee (eeano i?eueiioaioiino? (2) ?aeaae ne?i/aiii? eiaei??iino?. Ianooiia aeanoea?noue FC-aeaaa? E? ? iiaiei aiaeiaii a?aeiia?aeii? aeanoeaino? a?oi ? aeicaiey? ciaoiaeeoe iaiao?aei? ?aeaaee a iane?i/aiiiaei??ieo aeaaa?ao E?, ye? noiaie aeaaa? E? c? ne?i/aiiiaei??ieie eiiooaioaie. Eaia 2.2.12. sseui FC-aeaaa?a E? i?noeoue i?aeaeaaa?o A ne?i/aiii? eiaei??iino?, oi L i?noeoue ?aeaae ne?i/aiii? eiaei??iino?, yeee eaaeeoue a i?aeaeaaa?? A. Cie?aia c inoaiiuei? eaie, c a?aooaaiiyi ?acoeueoao?a i?i ?ica’yci?noue ne?i/aiiiaei??ieo aeaaa? E? oa?aeoa?enoeee (2, ui ?iceeaaeathoueny a noio aeaio nai?o i?eueiioaioieo i?aeaeaaa?, aeieeaa? oaea Oaa?aeaeaiiy 2.2.14. Iaoae L - aeaaa?a E? iaae aeia?eueiei iieai E, yea ?iceeaaea?oueny a noio L=A+B i?eueiioaioieo i?aeaeaaa? A ? A. sseui oi/a a iaeia ?c i?aeaeaaa? A aai A ne?i/aiiiaei??ia, oi aeaaa?a L iaeaea ?ica’ycia, a i?e charK(2 - ?ica’ycia. Aeiaaaeaiiy iniiaii? oai?aie oeueiai ia?aa?aoo aaaeaoueny oaeoe/ii ?iaeoeoe??th ca noiith ?ici??iinoae eiiooaio?a aeiaeaie?a ? i. a) ianooiii? eaie aeicaiey? aaaaeaoe a iiaeaeueoeo i??eoaaiiyo, ui ia?aoei aeiaeaie?a ? o?ea?aeueiei. Eaia 2.2.16. Iaoae aeaaa?a E? L ?iceeaaea?oueny a noio L=A+B i?aeaeaaa? A ? A c? ne?i/aiiiaei??ieie eiiooaioaie. Oiae? (a) A(A i?noeoueny a FC-oeaio?? FC(L) aeaaa?e L; (a) FC(L) - iaeaea ?ica’yciee ?aeaae aeaaa?e L; (a) yeui A2(B2=0, oi aoaeue-yeee ?aeaae aeaaa?e L, yeee ia? ioeueiaee ia?aoei c i?aei?inoi?ii A2+B2 ?ica’yciee. Eeth/iaei ?acoeueoaoii aeey iiooeo iao?ea?aeueieo FC-?aeaae?a a aeaaa?? E?, yea ?iceeaaea?oueny a noio aeaio nai?o i?aeaeaaa? c? ne?i/aiiiaei??ieie eiiooaioaie ? ianooiia Eaia 2.2.17. Iaoae L- aeaaa?a E?, ui ? noiith L=A+B nai?o i?aeaeaaa? A ? A c? ne?i/aiiiaei??ieie eiiooaioaie, yea caaeiaieueiy? ianooiiei oiiaai: (a) FC(L)=0; (a) aeaaa?a L i?noeoue oaeee FC-?aeaae N, ui L=N+A=N+B. Oiae? aeey aoaeue-yeiai aeaiaioa n0 ?c N aeaeo n0=a0+b0 c a0(Z(A), b0(Z(B) ni?aaaaeeeaa ??ai?noue [n0, N2]=0 c aeei?enoaiiyi yei? io?eiaii inoaii?e iaiao?aeiee ?acoeueoao aeey aeiaaaeaiiy oai?aie 2.2.20 Eaia 2.2.19. Iaoae L- aeaaa?a E?, yea ?iceeaaea?oueny a noio L=A+B i?aeaeaaa? A ? A c? ne?i/aiiiaei??ieie eiiooaioaie. sseui L i?noeoue oaeee FC-?aeaae N, ui L=A+N=B+N, oi L iaeaea ?ica’ycia. sse iane?aeie c oai?aie 2.2.20 c o?aooaaiiyi ?ica’yciino? noie aeaio ne?i/aiiiaei??ieo aeaaa? E? oa?aeoa?enoeee (2, io?eia?ii oaeee ?acoeueoao Oai?aia 2.2.22. Iaoae aeaaa?a E? iaae aeia?eueiei iieai oa?aeoa?enoeee (2 ?iceeaaea?oueny a noio L=A+B nai?o i?eueiioaioieo i?aeaeaaa? A ? A c? ne?i/aiiiaei??ieie eiiooaioaie. Oiae? aeaaa?a E? ?ica’ycia. Aeai?noeie (a aeayeiio nain?) aei aeaaa? E? c? ne?i/aiiiaei??ieie eiiooaioaie ? iaeaea aaaeaa? aeaaa?e E?, ye? oaeiae ? aeecueeeie aei aaaeaaeo aeaaa? E? (aeaaa?o E? iaae iieai aoaeaii iaceaaoe iaeaea aaaeaaith, yeui aiia i?noeoue aaaeaaee ?aeaae ne?i/aiii? eiaei??iino?). Iaeaea aaaeaa? aeaaa?e E? aaea ia iaia’yceiai ? FN-aeaaa?aie E? ? aeey ?o aea/aiiy aeei?enoiao?oueny ?io? i?aeoiaee. Ianooiia oai?aia ? iniiaiei ?acoeueoaoii o?aoueiai ia?aa?aoo ae?oaiai ?icae?eo: Oai?aia 2.3.1 Iaoae L - aeaaa?a E? iaae aeia?eueiei iieai, yea ?iceeaaea?oueny a noio L=A+B ne?i/aiiiaei??ii? iaae nai?i oeaio?ii i?aeaeaaa?e A ? iaeaea aaaeaai? i?aeaeaaa?e A. Oiae? aeaaa?a L iaeaea ?ica’ycia (oiaoi i?noeoue ?ica’yciee ?aeaae ne?i/aiii? eiaei??iino?). Cie?aia, ca?aene aeieeaa? oaea oaa?aeaeaiiy, yea ia? naiino?eia cia/aiiy Iane?aeie 2.3.2. Noia aaaeaai? oa iaeaea aaaeaai? aeaaa? E? iaeaea ?ica’ycia. Ine?eueee aeiaaaeaiiy oai?aie caiaeeoueny a e?ioe? e?ioe?a aei noie aeaio aaaeaaeo aeaaa? E?, oi i?e oeueiio ei?eniei ? ianooiiee ?acoeueoao, ui ocaaaeueith? oaa?aeaeaiiy i?i iaoaaaeaa?noue noie aeaio aaaeaaeo aeaaa? E?. Eaia 2.3.3. Iaoae L - aeaaa?a E?, M, N, - ?? aaaeaa? i?aeaeaaa?e, m1, m2(M, n1, n2(N - aeia?euei? aeaiaioe. Oiae?: yeui [m1, n2] ? [m2, n1] iaeaaeaoue noi? M+N, oi [[m1, n1], [m2, n2]]=0; yeui T - ?aeaae aeaaa?e L, yeee i?noeoueny a E-i?aei?inoi?? M+N, oi ?aeaae T ?ica’yciee ? nooi?iue eiai ?ica’yciino? ia ia?aaeuo? aeaio. Oae naii, ye ? a iiia?aaeiueiio ia?aa?ao?, aaaeeeao ?ieue i?e aea/aii? noi aeaio iaeaea aaaeaaeo aeaaa? E? a?ae?a?athoue FC-aeaaa?e E?. Oea iieaco? ianooiia oaa?aeaeaiiy: Eaia 2.3.5. Iaoae L - aeaaa?a E?, yea ?iceeaaea?oueny a noio L=A+B nai?o iaeaea aaaeaaeo i?aeaeaaa? A ? A. Oiae? FC-oeaio? I=FC(L) aeaaa?e L ? iaeaea ?ica’yciei ?aeaaeii aeaaa?e L. oa iane?aeie c iueiai: Iane?aeie 2.3.7. Iaoae L - aeaaa?a E?, yea ?iceeaaea?oueny a noio L=A+B nai?o iaeaea aaaeaaeo i?aeaeaaa? A ? A. Oiae? noia an?o iaeaea ?ica’ycieo ?aeaae?a aeaaa?e L ? iaeaea ?ica’yciei ?aeaaeii ?c L. Aeey aeiaaaeaiiy iniiaii? oai?aie o?aoueiai ia?aa?aoo aeaoaeueii aea/ai? aeanoeaino? i?i?iaeueiiai eiio?i?eeeaaeo aei oe??? oai?aie, yeee aaaaeaii a ?icaeyae ianooiiei icia/aiiyi: Icia/aiiy 2.3.11. Iaoae L - aeaaa?a E? iaae aeia?eueiei iieai, yea ?iceeaaea?oueny a noio L=A+B nai?o i?aeaeaaa? A ? A, aea A - ne?i/aiiiaei??ia iaae nai?i oeaio?ii aeaaa?a ? A iaeaea aaaeaaa. Aeaaa?o L aoaeaii iaceaaoe i?i?iaeueiei SF-eiio?i?eeeaaeii, yeui aiia caaeiaieueiy? ianooii? oiiae: L ia i?noeoue iaioeaaeo iaeaea ?ica’ycieo ?aeaae?a, cie?aia L ia ? iaeaea ?ica’yciith aeaaa?ith E?; i?aeaeaaa?a A ia i?noeoueny a a?eueo?e iaeaea aaaeaa?e i?aeaeaaa?? ?c L. Iniiai? aeanoeaino? i?i?iaeueieo SF-eiio?i?eeeaae?a aeathoueny aeaiia ianooiieie eaiaie: Eaia 2.3.14. Iaoae L=A+B - i?i?iaeueiee SF-eiio?i?eeeaae, aea A - ne?i/aiiiaei??ia iaae nai?i oeaio?ii i?aeaeaaa?a ?c L ? A iaeaea aaaeaaa i?aeaeaaa?a ?c L. sseui N - aaaeaaee ?aeaae ?c A c dim B/N<(, oi aeey aoaeue-yeiai aeaiaioa a(A, a(B i?aei?ino?? [a, N] iane?i/aiiiaei??iee iaae N. (aeey c?o/iino? i?aei?ino?? U aaeoi?iiai i?inoi?o V iaceaa?oueny ne?i/aiiiaei??iei iaae i?aei?inoi?ii W, yeui dimU+W/W<(). Eaia 2.3.16. Iaoae L=A+B - i?i?iaeueiee SF-eiio?i?eeeaae c dimA/Z(A)<(, dimB/N<( aeey aeayeiai aaaeaaiai ?aeaaea N ?c A ? iaoae N0={n(N| [A, n] ne?i/aiiiaei??iee iaae N}. Oiae? ([A, N](N)(N0, N0 - ?aeaae i?aeaeaaa?e A ? [A, N0](B. I?e aeiaaaeaii? iniiaii? oai?aie oeueiai ia?aa?aoo eeth/iao ?ieue c?a?aa oie oaeo, ui ye aoei aeiaaaeaii, ia?aoei [A, N](N iaioeueiaee ? oiio ca eaiith 2.3.16 iaioeueiaei aoaea ?aeaae N0 i?aeaeaaa?e A, yeee aea? ciiao iieacaoe, ui i?i?iaeueiee SF-eiio?i?eeeaae ia ?nio?. Inoaii?e 4-e ia?aa?ao 2-ai ?icae?eo aeena?oaoe?? i?noeoue ?yae ?acoeueoao?a, ui a?aeiinyoueny aei oae caaieo iio??eieo noi aeaaa? E? (ii aiaeia?? c oai???th a?oi aeaaa?o E? L aoaeaii iaceaaoe iio??eiith noiith nai?o i?aeaeaaa? A, A, N yeui L=A+B=A+C=B+C. Iio??ei? noie aeaaa? E? c’yaeythoueny i?e?iaei?i /eiii i?e aea/aii? ?aeaae?a aeaaa?e E?, yea ?iceeaaea?oueny a noio aeaio nai?o i?aeaeaaa?: yeui ? - ?aeaae aeaaa?e E? L, yea ?iceeaaea?oueny a noio L=A+B nai?o i?aeaeaaa? A ? A, oi ia?aoeie A?=A((?+A) ? A?=A((?+A) ? i?aeaeaaa?aie ?c A ? A a?aeiia?aeii, oaeeie, ui A?+ A?= A?+?= A?+? - ? i?aeaeaaa?ith ?c L, yea i?noeoue ?aeaae ? (oea aiaeia eaie Nanae?ia c oai??? a?oi). Aeey aea/aiiy iio??eieo noi aeaaa? E? a aeena?oaoe?ei?e ?iaio? aeei?enoiao?oueny i?aeo?ae, ?aeaeii aeecueeee aei oiai, yeee aoa cai?iiiiiaaiee ss.I.Nenaeii aeey aea/aiiy oaeoi?eciaaieo a?oi, a naia - caaaeaiiy ii/aoeiai? oaeoi?ecaoe?eii? caaea/? c oai??? a?oi aei aeayei? caaea/? c oai??? anioe?aoeaieo e?eaoeue, ?ica’ycaiiy yei? iaoiaeaie oai??? e?eaoeue aea? ciiao io?eiaoe a?aeiia?aei? oaa?aeaeaiiy i?i ii/aoeiao caaea/o. Ianooiia eaia noaa?aeaeo? oaeoe/ii, ui ca aeaiith iio??eiith noiith L aaaeaaeo aeaaa? E? iiaeia iiaoaeoaaoe anioe?aoeaio ne?i/aiiiaei??io aeaaa?o (iaai?ooth/o), i?e?aeiaia aei yei? aoea a ?ciii?oia L. Eaia 2.4.1. Iaoae L - aeaaa?a E? iaae aeia?eueiei iieai E, yea ?iceeaaea?oueny a iio??eio noio L=A+B=A+N=B+N, aea A, A - i?aeaeaaa?e ?c L ? N - aaaeaaee ?aeaae aeaaa?e L. sseui A(B= A(N= B(N=0, oi aeiaooie (() ia aaeoi?iiio i?inoi?? A iiaea aooe aecia/aiee oaeei /eiii, ui A0=A(() - e?i?eia aeaaa?a, yea caaeiaieueiy? oioiaei?noue (x,y,z)=(x,z,y), i?e?aeiaia aeaaa?a A0(-) ? aeaaa?ith E?, yea ?ciii?oia aeaaa?? E? A, a i?e?aeiaia aeaaa?a A*(-) ?ciii?oia aeaaa?? E? . (ooo (x,y,z) - anioe?aoi? aeaiaio?a x,y,z, a A* - iai?ai?yiee aeiaooie e?i?eieo aeaaa? A ? A+, a inoaii?e iiiaeaiiy ioeueiaa).C aeei?enoaiiyi oe??? eaie, ianooiiiai oaa?aeaeaiiy Eaia 2.4.3. Iaoae R - e?i?eia aeaaa?a, yea caaeiaieueiy? oioiaeiino? (x,y,z)=(x,z,y). sseui aeaaa?a R eiiooaoeaia, oi R oaeiae anioe?aoeaia ? oai?aie Iaeueoeaaa-Aaaeaea?aa?ia ?c oai??? anioe?aoeaieo aeaaa? anoaiiaeth?oueny aoaeiaa iio??eieo noi aaaeaaeo aeaaa? E?: Oai?aia 2.4.5. Iaoae L - iai?eueiioaioia ne?i/aiiiaei??ia aeaaa?a E? iaae aeaaa?a?/ii caieiaiei iieai, yea ?iceeaaea?oueny a iio??eio noio L=A+B=A+N=B+N aaaeaaeo i?aeaeaaa? A, A ? aaaeaaiai ?aeaaeo N, oaeeo, ui A(B=A(N=B(N=0. Oiae? aeaaa?a L i?noeoue i?eueiioaioiee ?aeaae I (iiaeeeai, I=0), yeee ?iceeaaea?oueny a iio??eio noio I=AI+BI= AI+NI= BI+NI aeayeeo i?aeaeaaa? AI(A, BI(B ? ?aeaaeo NI(N ? aeayeo i?aeaeaaa?o D, yea ?iceeaaea?oueny a i?yiee aeiaooie iaaaaeaaeo aeaiaei??ieo aeaaa? E? ? i?e oeueiio L=D+I, D(I=0. O?ao?e ?icae?e aeena?oaoe?eii? ?iaioe i?ena’y/aiee aea/aiith noi anioe?aoeaieo aeaaa? ? e?eaoeue, ye? caaeiaieueiythoue aeayeei oiiaai, ui aeecuee? aei i?eueiioaioiino? /e eiiooaoeaiino?. A ia?oiio ia?aa?ao? oeueiai ?icae?eo ?icaeyaeathoueny anioe?aoeai? (iane?i/aiiiaei??i?) aeaaa?e iaae aeia?eueiei iieai, ye? ?iceeaaeathoueny a noio aeaio nai?o iaeaea eiiooaoeaieo i?aeaeaaa? (oiaoi, i?aeaeaaa?, ui i?noyoue eiiooaoeaiee ?aeaae ne?i/aiii? eiaei??iino?). Iaeaea eiiooaoeai? aeaaa?e ? i?e?iaei?i ocaaaeueiaiiyi eiiooaoeaieo aeaaa?, caaeiaieueiythoue aeayeei iie?iii?aeueiei oioiaeiinoyi (oiaoi ? ??-aeaaa?aie) ? aea/aiiy ?o noi ? ianooiiei e?ieii i?ney aeine?aeaeaiiy noi eiiooaoeaieo anioe?aoeaieo e?eaoeue, yea aoei i?iaaaeaii ca inoaii? ?iee a ?iaioao TH.A.Aaooo??ia, A.Aeaeaia?oii ? I.Eaaaey. Iniiaiei ?acoeueoaoii, yeee io?eiaiee a oeueiio iai?yi? a aeena?oaoe?ei?e ?iaio? ? ianooiia oaa?aeaeaiiy: Oai?aia 3.1.1. Iaoae A - anioe?aoeaia aeaaa?a iaae aeia?eueiei iieai, yea ?iceeaaea?oueny a noio A=A+N nai?o iaeaea eiiooaoeaieo i?aeaeaaa? A ? N. Oiae? aeaaa?a A i?noeoue i?eueiioaioiee ?aeaae ? oaeee, ui oaeoi?-aeaaa?a A/? iaeaea eiiooaoeaia. Aeacai? aeaye? i?ino? aeanoeaino? iaeaea eiiooaoeaieo anioe?aoeaieo aeaaa?, ye? oe?iei aeei?enoiaothoueny aeey aeiaaaeaiiy iniiaii? oai?aie oeueiai ia?aa?aoo: Eaia 3.1.6. Iaoae A - iaeaea eiiooaoeaia anioe?aoeaia aeaaa?a ? ? - eiiooaoeaiee ?aeaae aeaaa?e A c dimA/I<(. Oiae?: [A, A]I i?noeoueny a aeayeiio i?eueiioaioiiio ?aeaae? aeaaa?e A; aeey aeayeiai i?eueiioaioiiai ?aeaaeo J oaeoi?-aeaaa?a A/J i?noeoue ne?i/aiiiaei??iee ?aeaae T/J, oaeee, oaeoi?-aeaaa?a A/T eiiooaoeaia. I?e aeine?aeaeaii? noi iaeaea eiiooaoeaieo anioe?aoeaieo aeaaa? aoee aea/ai? aeanoeaino? anioe?aoeaieo aeaaa?, ye? i?noyoue i?eueiioaioiee ?aeaae, oaeoi?-aeaaa?a ca yeei iaeaea eiiooaoeaia (aeey c?o/iino? aiie iaceaathoueny a aeena?oaoe?ei?e ?iaio? NCF-aeaaa?aie). Aeyaeeinue, ui oae? aeaaa?e iathoue ?yae aaaeeeaeo aeanoeainoae, ye? aea? ianooiia Oaa?aeaeaiiy 3.1.8. Ni?aaaaeeea? ianooii? oaa?aeaeaiiy: eiaeia i?aeaeaaa?a ? eiaeia oaeoi?-aeaaa?a NCF-aeaaa?e ? oaeiae NCF-aeaaa?aie; yeui A ? A ? NCF-aeaaa?aie, oi ?o i?yiee aeiaooie A(A oaeiae ? NCF-aeaaa?ith: eiaeia ?icoe?aiiy NCF-aeaaa?e ca aeiiiiiaith NCF-aeaaa?e ciiao ? NCF-aeaaa?ith, aieiai? c yeeo - caieiai?noue eeano NCF-aeaaa? a?aeiinii ne?i/aiieo i?yieo aeiaooe?a oa caieiai?noue a?aeiinii ?icoe?aiue aeicaieeea caanoe caaea/o aea/aiiy noi iaeaea eiiooaoeaieo aeaaa? aei aea/aiiy noi iaeaea i?eueiioaioieo anioe?aoeaieo aeaaa? (oiaoi, oaeeo aeaaa?, ye? i?noyoue i?eueiioaioiee ?aeaae ne?i/aiii? eiaei??iino?). Aeey oeueiai aeaoaeueii aoa aea/aiee i?i?iaeueiee eiio?i?eeeaae aei oaa?aeaeaiiy oai?aie, yeee aeey c?o/iino? aoa iacaaiee AI-eiio?i?eeeaaeii: Icia/aiiy 3.1.13. Anioe?aoeaio aeaaa?o A iaae aeia?eueiei iieai F, yea ?iceeaaea?oueny a noio A=A+N aeaio iaeaea eiiooaoeaieo i?aeaeaaa? A ? N aoaeaii iaceaaoe i?i?iaeueiei AI-eiio?i?eeeaaeii, yeui A caaeiaieueiy? ianooiiei oiiaai: A ia ? NCF-aeaaa?ith; i?aeaeaaa?e A ? N i?noyoue eiiooaoeai? ?aeaaee A0 ? a?aeiia?aeii N0 oae?, ui dim B/B0+dim C/C0<( ? /enei dimA/(B0+C0) iaeiaioa; aeaaa?a A ia i?noeoue iaioeaaeo ?aeaae?a, ye? eaaeaoue a F-i?aei?inoi?? B0+C0 ?c oiiae 2). Oe?eee ?yae eai i?ena’y/aii aea/aiith i?i?iaeueiiai AI-eiio?i?eeeaaeo (oaa?aeaeaiiy iniiaii? oai?aie oeueiai ia?aa?aoo aea?aaeaioia oiio, ui i?i?iaeueiiai AI-eiio?i?eeeaaeo ia ?nio?). Cie?aia, c aeei?enoaiiyi aeanoeainoae NCF-aeaaa? anoaiiaeaii oaea oaa?aeaeaiiy: Eaia 3.1.14. Iaoae A=A+N - i?i?iaeueiee AI-eiio?i?eeeaae. Oiae? aeey eiaeiiai iaioeueiaiai ?aeaaea ? aeaaa?e A oaeoi?-aeaaa?a A/? ? NCF-aeaaa?ith. E??i oiai, aeaaa?a A ia i?noeoue iaioeueiaeo NCF-?aeaae?a. Ine?eueee ia iaeiiio c e?ie?a aeiaaaeaiiy iniiaii? oai?aie oeueiai ia?aa?aoo c’yaeythoueny noie iaeaea i?eueiioaioieo anioe?aoeaieo aeaaa? (anioe?aoeaio aeaaa?o iaae iieai aoaeaii iaceaaoe iaeaea i?eueiioaioiith, yeui aiia i?noeoue i?eueiioaioiee ?aeaae ne?i/aiii? eiaei??iino?), oi aeae?eueea aeiiii?aeieo oaa?aeaeaiue noinothoueny aea/aiiy aeanoeainoae oaeeo aeaaa?. Eaia 3.1.19. sseui ? - i?aaee (e?aee) iaeaea i?eueiioaioiee ?aeaae aeaaa?e A, oi A i?noeoue aeayeee i?eueiioaioiee ?aeaae J oaeee, ui I+J/J - ne?i/aiiiaei??iee i?aaee (a?aeiia?aeii e?aee) ?aeaae aeaaa?e A. Ianooii? aeaa iane?aeee c aeuacaaaeaieo eai iieacothoue, ui iaeaea i?eueiioaioi? anioe?aoeai? aeaaa?e iathoue o? ae iniiai? aeanoeaino?, ui ? i?eueiioaioi?: Iane?aeie 3.1.20 Iaoae A - anioe?aoeaia aeaaa?a ? ? - i?aaee (e?aee) iaeaea i?eueiioaioiee ?aeaae aeaaa?e A. Oiae? ? i?noeoueny a aeayeiio iaeaea i?eueiioaioiiio ?aeaae? aeaaa?e A. Iane?aeie 3.1.21. sseui anioe?aoeaia aeaaa?a A i?noeoue iaeaea i?eueiioaioiee ?aeaae ? c iaeaea i?eueiioaioiith oaeoi?-aeaaa?ith A/?, oi aeaaa?a A iaeaea i?eueiioaioia. Ianooiiee ?acoeueoao, i?e aeiaaaeaii? yeiai aeei?enoiaooueny aeuaiaaaaeai? aeanoeaino? iaeaea i?eueiioaioieo aeaaa?, ? ocaaaeueiaiiyi aeey anioe?aoeaieo aeaaa? iaae iieai a?aeiii? oai?aie I.Eaaaey i?i i?eueiioaioi?noue anioe?aoeaiiai e?eueoey, yea ?iceeaaea?oueny a noio aeaio nai?o i?eueiioaioieo i?aee?eaoeue. Oaa?aeaeaiiy 3.1.22. sseui anioe?aoeaia aeaaa?a iaae aeia?eueiei iieai ?iceeaaea?oueny a noio A=A+N c iaeaea i?eueiioaioieie i?aeaeaaa?aie A ? N, oi aeaaa?a A iaeaea i?eueiioaioia. Oea oaa?aeaeaiiy aeei?enoao?oueny i?e aeiaaaeaii? oai?aie 3.1.1, a oaeiae iiaea i?aaenoaaeyoe naiino?eiee ?ioa?an. Ua iaeiei eeanii anioe?aoeaieo aeaaa? (oa e?eaoeue), ye? aeecuee? aei eiiooaoeaieo ? e??ai i?eueiioaioi? aeaaa?e (aeaaa?e, i?e?aeiai? aei yeeo ? i?eueiioaioieie aeaaa?aie E?). Noie e??ai i?eueiioaioieo anioe?aoeaieo aeaaa? oa e?eaoeue aea/athoueny a ia?aa?ao? 3.2. Aeey anioe?aoeaieo aeaaa? (? e?eaoeue) iiaeia noi?ioethaaoe aiaeia i?iaeaie Eaaaey: /e aoaea e??ai ?ica’yciith anioe?aoeaia aeaaa?a, ui ? noiith aeaio e??ai i?eueiioaioieo i?aeaeaaa?? C?icoi?ei, ui a ca’yceo c ?acoeueoaoaie, a?aecia/aiei a ?icae?e? 2 aeaii? ?iaioe, oea ieoaiiy ? oe?eaaei o?eueee aeey iane?i/aiiiaei??ieo anioe?aoeaieo aeaaa? oa aeey e?eaoeue. Ine?eueee c iaeyaeo ia a?aeii? ?acoeueoaoe Aeaeaii?iana e??ai i?eueiioaioia e?eueoea i?noeoue i?eueiioaioiee ?aeaae , oaeoi?-e?eueoea ca yeei e??ai i?eueiioaioia eeano 2, oi a aeena?oaoe?ei?e ?iaio? ie?aii aeae?ey?oueny aeiaaeie noie eiiooaoeaiiai oa e??ai i?eueiioaioiiai eeano 2 e?eaoeue. I?e aea/aii? noi e??ai i?eueiioaioieo anioe?aoeaieo e?eaoeue oaeiae aeei?enoiao?oueny aeayea eiiooaoi?ia /eneaiiy, yea ?icaeiooa a ?yae? eai. Ianooiia eaia iieaco?, ui aeey anioe?aoeaieo e?eaoeue a aeayeeo aeiaaeeao iiaeia aeacaoe aeanoeaino?, ui noiae? ia aeanoeaino? noi aeaaa? oa e?eaoeue E?. Eaia 3.2.1. Iaoae R - anioe?aoeaia e?eueoea, yea ?iceeaaea?oueny a noio R=A+B eiiooaoeaiiai i?aee?eueoey A ? aeayeiai i?aee?eueoey A. Oiae? ni?aaaaeeea? aeeth/aiiy [A, Z(B)][A, Z(B)]([B, B] Oey eaia aea? iiaeeea?noue aoaeoaaoe i?eueiioaioi? ?aeaaee a i?aee?eueoe? A (oea iani?aaae? aoaea aeiaooie [A, Z(B)][A, Z(B)]), a ca ieie e??ai ?icayci? i?aeaeaaa?e, ye oea iieaco? ianooiia eaia. Eaia 3.2.4. Iaoae R - anioe?aoeaia e?eueoea, yea ?iceeaaea?oueny a noio R=A+B, aea A - eiiooaoeaia i?aee?eueoea ?c R ? A - e??ai i?eueiioaoia i?aee?eueoea eeano i?eueiioaioiino? n. Oiae? J=[A, Z(B)]+[B,B] - e??ai ?icaycia i?aee?eueoea a e?eueoe? E? R ? nooi?iue ?icayciio? J ia ia?aaeuo? n+[log2(n)], aea [x] - oe?ea /anoeia ae?eniiai /enea . C aeei?enoaiiyi aeayeeo i?aeoiae?a ? eiiooaoi?iiai /eneaiiy c oai??? aeaaa? E? aeine?aeaeai? anioe?aoeai? e?eueoey, ye? ? noiaie eiiooaoeaiiai i?aee?eueoey oa e??ai i?eueiioaioiiai eeano 2 i?aee?eueoey. Anoaiiaeaii, ui aiie e??ai ?ica’yci? nooiaiy ia a?eueoa i?ae 6. Aeey aeiaaaeaiiy oai?aie 3.2.9, yea ? iniiaiei ?acoeueoaoii o?aoueiai ia?aa?aoo anoaiiaeaii ua iaeio aeanoea?noue noie eiiooaoeaiiai oa aeia?eueiiai e?eueoey, yea iiae?aia aiaeia?/i?e aeanoeaino? aeaaa? E?. Eaia 3.2.8. Iaoae R - anioe?aoeaia e?eueoea, yea ?iceeaaea?oueny a noio R=A+B, aea A - eiiooaoeaia i?aee?eueoea ?c R ? A - aeia?eueia i?aee?eueoea ?c R. Oiae? aeeiio?oueny ni?aa?aeiioaiiy [A, Bi][A, Bj]([B,B]+ABj+1+Bi+1A+ABi+jA aea Bi=[B,B,B…B] (ooo ? ?ac iiaoi?th?oueny A). A ianooii?e oai?ai?, e??i aeiaaaeaiiy e??ai? ?ica’yciino?, aaaeeeaith ? oaeiae io?eiaia ioe?iea nooiaiy e??ai? ?ica’yciino? a caeaaeiino? a?ae eeano e??ai? i?eueiioaioiino? e??ai i?eueiioaioiiai aeiaeaieo. Aeey aeaaa? E? oaei? ioe?iee iiee ui iaia? ? i?aeoiaee c oai??? anioe?aoeaieo e?eaoeue iiaeooue aooe ei?enieie a oai??? aeaaa? E?. Oai?aia 3.2.9. Iaoae R - anioe?aoeaia e?eueoea, yea ?iceeaaea?oueny a noio R=A+B eiiooaoeaiiai i?aee?eueoey A ? e??ai i?eueiioaioiiai i?aee?eueoey A eeano i?eueiioaioiino? n. Oiae? e?eueoea R e??ai ?ica’ycia nooiaiy ia aeuiai i?ae (n+1)(n+2)/2. A /aoaa?oiio ?icae?e? aeena?oaoe?eii? ?iaioe aea/athoueny aeaaa?e E?, a yeeo aeiiiaithaai? an? i?aeaeaaa?e oa an? iaeiiaei??i? i?aeaeaaa?e, a oaeiae a?oie c aeayeeie nenoaiaie aeiiiaithaaieo iiaea?oi. Ooo ii aiaeia?? c oai???th a?oi i?aeaeaaa?o A aeaaa?e L aoaeaii iaceaaoe aeiiiaithaaiith, yeui a L ?nio? i?aeaeaaa?a A oaea, ui L=A+B, A(B=0. Aea/aiiy ne?i/aiieo a?oi, a yeeo aeiiiaithaai? an? i?aea?oie ?icii/aa O.Oiee ua a o?eaeoeyoeo ?ieao, a iiaiee iien oaeeo a?oi io?eiaii a ?iaio?13 I.A.*a?i?eiai?. Ianooiia oaa?aeaeaiiy ca’yco? i?ae niaith oiiae aeiiiaithaaiino? c oiiaith aeinoaoiuei? e?eueeino? i?aeaeaaa? eiaei??iino? 1. Eaia 4.1.3. Iaoae L iaioeueiaa ne?i/aiiiaei??ia aeaaa?a E? iaae aeia?eueiei iieai. Oiae? ianooii? oiiae aea?aaeaioi?: a aeaaa?? L aeiiiaithaaia eiaeia i?aeaeaaa?a; a L aeiiiaithaaia eiaeia iaeiiaei??ia i?aeaeaaa?a; ia?aoei (1(L) an?o i?aeaeaaa? eiaei??iino? 1 a L aei??aith? 0. Aeey c?o/iino? aeaaa?e E? c aeiiiaithaaieie i?aeaeaaa?aie a aeena?oaoe?ei?e ?iaio? iaceaathoueny n-aeaaa?aie. C eaie 4.1.3 aeieeaathoue iane?aeee, ye? oe?iei aeei?enoiaothoueny a oeueiio ia?aa?ao?: Iane?aeie 4.1.4. I?yia noia aeaio ne?i/aiiiaei??ieo n-aeaaa? E? ? ciiao n-aeaaa?ith E?. Ianooiiee iane?aeie aeicaiey? ?icoe?yoe iiea eiao?oe??io?a aeaaa?e E? c aeiiiaithaaieie i?aeaeaaa?aie. . A oai?ai? 4.1.6 - iaeiiio c iniiaieo ?acoeueoao?a ia?aa?aoo 4.1 aeaii iien ne?i/aiiiaei??ieo aeaaa? E? c aeiiiaithaaieie i?aeaeaaa?aie iaae aeia?eueiei aeineiiaeei iieai oa?aeoa?enoeee (2 ? iaae aeia?eueiei iieai oa?aeoa?enoeee ?=2. Oai?aia 4.1.6. A ne?i/aiiiaei??i?e aeaaa?? E? L iaae aeia?eueiei iieai oa?aeoa?enoeee (2 oiae? ? o?eueee oiae? aeiiiaithaai? an? i?aeaeaaa?e, eiee L ?iceeaaea?oueny a i?yio noio L=A(B ?aeaae?a A ? A, aea A aai aei??aith? ioeth, aai i?yia noia o?eaei??ieo i?inoeo aeaaa? E? oeio A1, A=A1(A2 - iai?ai?yiee aeiaooie aaaeaaeo i?aeaeaaa? E? A1 ? A2, i?e/iio A1 - i?yia noia iaeiiaei??ieo ?aeaae?a aeaaa?e L; a ne?i/aiiiaei??i?e aeaaa?? E? L iaae aeia?eueiei iieai oa?aeoa?enoeee ?=2 oiae? ? o?eueee oiae? aeiiiaithaai? an? i?aeaeaaa?e, eiee L ?icaycia aeaeo L=B=B1(B2 c ioieoo 1 oe??? oai?aie. Ia?oei e?ieii i?e aeiaaaeaii? oe??? oai?aie ? anoaiiaeaiiy aoaeiae ?ica’ycii? ne?i/aiiiaei??ii? aeaaa?e E? c aeiiiaithaaieie i?aeaeaaa?aie. Eaia 4.1.9. Iaoae L - iaioeueiaa ne?i/aiiiaei??ia ?ica’ycia aeaaa?a E? iaae aeia?eueiei iieai. Aeaaa?a E? L oiae? ? o?eueee oiae? ? n-aeaaa?ith, eiee L=A(B, aea A - aaaeaaa i?aeaeaaa?a ?c L, A - aaaeaaee ?aeaae ?aeaae ?c L, yeee ?iceeaaea?oueny a i?yio noio iaeiiaei??ieo ?aeaae?a aeaaa?e L. Ianooiiei e?ieii ? aea/aiiy i?inoeo n-aeaaa? E?. I?e aeine?aeaeaii? oaeeo aeaaa? E? aeei?enoiaothoueny ?acoeueoaoe A.N.Aeaeoiaae?eueaea?aa i?i aoaeiao i?inoeo aeaaa? E? iaae aeineiiaeeie iieyie, ye? iathoue i?aeaeaaa?o eiaei??iino? 1. Eaia 4.1.11. sseui L - ne?i/aiiiaei??ia i?inoa aeaaa?a E? iaae aeineiiaeei iieai, a ye?e aeiiiaithaai? an? i?aeaeaaa?e, oi L - ?icuaiethaaia o?eaei??ia i?inoa (oeio A1) aeaaa?a. C aeei?enoaiiyi oe??? eaie aeiaiaeeoueny ianooiia oaa?aeaeaiiy, yea ie?aii aeae?ey? n-aeaaa?e E? oa?aeoa?enoeee ?=2. Eaia 4.1.13. Ne?i/aiiiaei??ia n-aeaaa?a E? iaae aeia?eueiei iieai E oa?aeoa?enoeee ?=2 ?ica’ycia. ?acii eaiith 4.1.9 oea aea? aaea iien ne?i/aiiiaei??ieo aeaaa? E? c aeiiiaithaaieie i?aeaeaaa?aie iaae aeia?eueiei iieai oa?aeoa?enoeee ?=2. Aeiaaeie iiceoeaii? iaia?ii? oa?aeoa?enoeee aeyaeany aaae/ei ? iniiaia oai?aia aea? iien aeaaa? E? c aeiiiaithaaieie i?aeaeaaa?aie o?eueee iaae aeineiiaeeie iieyie, oi/a i?e oiia? iieno i?inoeo n-aeaaa? E? iaae aeia?eueieie iieyie aiia iiaea aooe iioe?aia ia aeia?euei? iiey. A?aecia/eii, ui o aeiaaeeo aeaaa? E? ioeueiai? oa?aeoa?enoeee iien oaeeo aeaaa? E? (ye aeaaa? E? c ioeueiaei ia?aoeiii an?o i?aeaeaaa? eiaei??iino? 1) i?noeoueny a ?iaio? E.Aioiaia, oi/a a ?io?e oi?i? eiai iiaeia io?eiaoe c ?ia?o AE.O?ona aai ?ia?o c oai??? aeaaa? E? Oo?ooe?iaa?a oa Ae.Oaoa?na, i?ena’y/aieo aea/aiith aeaaa? E? c ioeueiaith i?aeaeaaa?ith O?aoo?i?. Aeey iane?i/aiiiaei??ieo aeaaa? E? iiaeia oaeiae ?icaeyaeaoe ieoaiiy i?i aoaeiao oaeeo aeaaa? ca oiiae aeiiiaithaaiino? a ieo an?o iaeiiaei??ieo i?aeaeaaa?. Eieaeueii ?ica’yci? aeaaa?e E? oaeiai ?iaeo ? i?inoi i?aeaeaaa?aie aeaea?oiai? noie aeaiaei??ieo iaaaaeaaeo aeaaa? E?: Eaia 4.2.3. A iaioeueia?e eieaeueii ?ica’yci?e aeaaa?? E? L iaae aeia?eueiei iieai oiae? ? o?eueee oiae? aeiiiaithaai? an? iaeiiaei??i? i?aeaeaaa?e, eiee L ?ciii?oii aeeaaea?oueny a aeaea?oiao noio aeaiaei??ieo iaaaaeaaeo aeaaa? E?. A iane?i/aiiiaei??iiio aeiaaeeo oiiaa aeiiiaithaaiino? an?o iaeiiaei??ieo i?aeaeaaa? aaea ia ? aea?aaeaioiith oiia? aeiiiaithaaiino? an?o i?aeaeaaa?. A?aeiia?aeiee i?eeeaae, yeee iiaoaeiaaii a eai? 4.2.5 ? i?aeaeaaa?ith aeaea?oiai? noie aeaiaei??ieo iaaaaeaaeo aeaaa? E?, yea i?noeoue iaaeiiiaithaai? iane?i/aiiiaei??i? i?aeaeaaa?e ? iiaea aaaaeaoeny aiaeiaii a?aeiia?aeiiai i?eeeaaeo TH.I.Ai?/aeiaa c oai??? a?oi, a naia ia??iaee/ii? a?oie, a ye?e aeiiiaithaai? an? oeeee?/i? i?eia?i? i?aea?oie, aea yea i?noeoue iaaeiiiaithaai? iane?i/aii? i?aea?oie. Ianooiia oai?aia ? aiaeiaii aeey aeaaa? E? a?aeiia?aeii? oai?aie I.A.*a?i?eiai? c oai??? a?oi i?i aoaeiao a?oi c aeiiiaithaaieie i?aea?oiaie: Oai?aia 4.2.4. A eieaeueii ?ica’yci?e aeaaa?? L(0 iaae aeia?eueiei iieai oiae? ? o?eueee oiae? aeiiiaithaai? an? i?aeaeaaa?e, eiee aiia ?iceeaaea?oueny a a iai?ai?yiee aeiaooie L=A(B, aea A - aaaeaaee ?aeaae ?c L, yeee ?iceeaaea?oueny a i?yio noio iaeiiaei??ieo ?aeaae?a aeaaa?e L, B - aaaeaaa i?aeaeaaa?a ?c L. I?e ia?aoiae? a?ae eieaeueii ?ica’ycieo aei aeia?eueieo aeaaa? E? c aeiiiaithaaieie iaeiiaei??ieie i?aeaeaaa?aie aeieea? ieoaiiy i?i eieaeueio ne?i/aii?noue. Oi/a a aeaaa?ao E? c aeiiiaithaaieie iaeiiaei??ieie i?aeaeaaa?aie ? “aaaaoi” i?aeaeaaa? ne?i/aiii? eiaei??iino?, oea ia i?ioeaaao aiaeia?/i?e neooaoe?? a oai??? a?oi, ia aa?aioo? iaa?oue iai?inoioe aeaaa?e E?. Oiio i?e aea/aii? iane?i/aiiiaei??ieo aeaaa? E? c aeiiiaithaaieie iaeiiaei??ieie i?aeaeaaa?aie ia aeaaa?o E? iaeeaaea?oueny oiiaa eieaeueii? ne?i/aiiino?, a o aeiaaeeo iniiaiiai iiey oa?aeoa?enoeee (2 ua e oiiae ce?/aiiino? aaceno aeaaa?e. Oai?aia 4.2.6. A eieaeueii ne?i/aiiiaei??i?e aeaaa?? E? L c? ce?/aiiei aacenii iaae aeineiiaeei iieai oa?aeoa?enoeee (2 oiae? ? o?eueee oiae? aeiiiaithaai? an? iaeiiaei??i? i?aeaeaaa?e, eiee L ?ciii?oii aeeaaea?oueny a aeaea?oiao noio o?eaei??ieo i?inoeo aeaaa? E? oeio A1. Oae naii ye ? aeey ne?i/aiiiaei??ieo aeaaa? E? iniaeeaei ? aeiaaeie iiey oa?aeoa?enoeee ?=2. Eieaeueii ne?i/aii? aeaaa?e E? c aeiiiaithaaieie i?aeaeaaa?aie a oeueiio aeiaaeeo eieaeueii ?ica’yci? ? oiio ?o aoaeiaa caiaeeoueny aei ?icaeyiooiai ?ai?oa aeiaaeeo. Oai?aia 4.2.7. A eieaeueii ne?i/aiiiaei??i?e aeaaa?? E? L aeia?eueiei iieai oa?aeoa?enoeee ?=2 oiae? ? o?eueee oiae? aeiiiaithaai? an? iaeiiaei??i? i?aeaeaaa?e, eiee L ?ciii?oii aeeaaea?oueny a aeaea?oiao noio aeaiaei??ieo iaaaaeaaeo aeaaa? (cie?aia, L ?ica’ycia nooiaiy (2). I?e aeiaaaeaii? oe??? oai?aie nooo?ai aeei?enoiaoaany iien ne?i/aiiiaei??ieo aeaaa? E? oa?aeoa?enoeee ?=2, a yeeo aeiiiaithaai? an? i?aeaeaaa?e, io?eiaiee a iiia?aaeiueiio ia?aa?ao?. A ianooiiiio o?aoueiio ia?aa?ao? oeueiai ?icae?eo aea/athoueny aaea a?oie c ??cieie nenoaiaie aeiiiaithaaieo i?aea?oi. A ?aeaeiiio a?aeiioaii? ? aaaaoi ni?eueieo ?en ye a i?aeoiaeao, ui aeei?enoiaothoueny i?e aea/aii? aeaaa? E? c ??cieie nenoaiaie aeiiiaithaaieo i?aeaeaaa?, oae ? a?oi c oeie /e ?ioeie nenoaiaie aeiiiaithaaieo i?aea?oi. A oeueiio ia?aa?ao? anoaiiaeaii aoaeiao iane?i/aiieo iai?eia?ieo eieaeueii nooi?i/anoeo a?oi c aeiiiaithaaiie iai?eia?ieie iaoeeee?/ieie i?aea?oiaie ? ioa?aeoa?eciaaii ne?i/aii? a?oie c aeiiiaithaaieie iaaenee?anueeeie i?aea?oiaie (i?e aeayeeo iaiaaeaiiyo). Cie?aia, aeyaeeinue, ui eieaeueii nooi?i/ano? a?oie c aeiiiaithaaieie iai?eia?ieie iaoeeee?/ieie i?aea?oiaie iaeaaeeoue iaeiiio c eean?a a?oi, ye? aoee aea/ai? ?ai?oa ?ioeie aaoi?aie (a?oie c aeiiiaithaaieie i?aea?oiaie - oe?eeii oaeoi?eciaai? a?oie - iienai? I.A.*a?i?eiaith, c aeiiiaithaaieie iaoeeee?/ieie i?aea?oiaie - I.I.Coa). Oai?aia 4.3.5. A iane?i/aii?e eieaeueii nooi?i/ano?e a?oi? G oiae? ? o?eueee oiae? aeiiiaithaai? an? iai?eia?i? iaoeeee?/i? i?aea?oie, eiee aeeiio?oueny iaeia c oiia: G - eieaeueii ne?i/aiia ? a i?e aeiiiaithaai? an? ?? i?aea?oie; G - eieaeueii ne?i/aiia ? a i?e aeiiiaithaai? an? ?? iaoeeee?/i? i?aea?oie; G - iane?i/aiia oeeee?/ia a?oia. I?e aeiaaaeaii? oe??? oai?aie aeei?enoiaoaaeenue ?acoeueoaoe I.I.Coa i?i aoaeiao a?oi c aeiiiaithaaieie iaoeeee?/ieie i?aea?oiaie ? i?ney ?yaeo e?ie?a aeiaaaeaiiy caiaeeoueny aei ia?aa??ee ia aeiiiaithaai?noue aeayeeo nenoai i?aea?oi a aaeeoeai?e a?oi? ?aoe?iiaeueieo /enae Q+. Ieoaiiy i?i no?oeoo?o ne?i/aiieo a?oi c aeiiiaithaaieie iaaenee?anueeeie i?aea?oiaie (oiaoi i?aea?oiaie ye? i?noyoue oi/a a iaeio iaiaeeie/io neeianueeo i?aea?oio an??? a?oie) aeieeei a ca’yceo c a?aeiiith oai?aiith O.Oieea i?i oa, ui a ne?i/aii?e a?oi? oiae? ? o?eueee oiae? aeiiiaithaai? an? neeianuee? i?aea?oie, eiee aiia ?ica’ycia. ?icoe?eaoe nenoaio aeiiiaithaaieo i?aea?oi, ie caoaeo?ii eean a?oi, ui i?aeeyaathoue aea/aiith. Aoaeiao ne?i/aiieo a?oi nooiaiy ?ica’yciino? 3 c aeiiiaithaaieie iaaenee?anueeeie i?aea?oiaie aea? ianooiia oai?aia (aeainooiaiaai ?ica’yci? a?oie oaeiai ?iaeo aoee aea/ai? ?ai?oa A?oie c aeiiiaithaaieie iaaenee?anueeeie i?aea?oiaie iaceaathoueny aeey c?o/iino? UH-a?oiaie. Oai?aia 4.3.12. Iaoae G - ne?i/aiia o?enooiaiaai ?ica’ycia a?oia, ii?yaeie yei? ae?eeoueny ia iaioa i?ae ia o?e ??cieo i?inoeo /enea. A?oia G oiae? ? o?eueee oiae? ? UH-a?oiith, eiee aiia i?aaenoaaey?oueny iai?ai?yiei aeiaooeii G=A(B i?aea?oi A ? A, ye? caaeaieueiythoue ianooiiei oiiaai: i?aea?oia A ?iceeaaea?oueny a i?yiee aeiaooie aeayeeo iaoeeee?/ieo i?i?iaeueieo ii?iaeueieo i?aea?oi A1, …, Ak (k(1)$ i?aea?oia A iaaaaeaaa oe?eeii oaeoi?eciaaia; aeey eiaeiiai ? (1(i(k) oeaio?ae?caoi? CG(Ai) ia i?noeoue aeiaeii? neeianueei? i?aea?oie a?oie G, ii?yaeie yei? aca?iii i?inoee c ii?yaeeii a?oie Ai; aeey eiaeiiai ? (1(i(k) a?oia Ai(Di , aea Di - aeiiiaiaiiy aei oeaio?ae?caoi?a CB(Ai) a B, ? aai a?oiith oeio 1), 2) aai 3) c oaa?aeaeaiiy 4.3.7 aai c a?oiith c eaie 4.3.11. oi/a a iaeia c i?aea?oi Di (1(i(k) iaaaaeaaa. A oaa?aeaeaii? 4.3.7 aeacaii oaeoe/ii “eieaeueio” aoaeiao aeuacaaaeaieo a?oi Oaa?aeaeaiiy 4.3.7. Iaoae G - ne?i/aiia iai?eia?ia a?oia c aaaeaaeie neeianueeeie i?aea?oiaie, i?aea?oia O?oo?iaa F yei? ? i?i?iaeueiith ii?iaeueiith i?aea?oiith a?oie G. A?oia G oiae? ? o?eueee oiae? ? UH-a?oiith, eiee aiia iaeaaeeoue iaeiiio ?c ianooiieo eean?a a?oi (ne??cue F - neeianueea aeaiaioa?ia aaaeaaa ?-i?aea?oia a?oie G): G=F((b(, aea (b( - oe?eeii oaeoi?eciaaia a?oia, eiaeia i?aea?oia yei? ae?? iaca?aeii ia F; G=F((Q(R), aea Q ? R - neeianuee? i?aea?oie a?oie G, i?eia?i? a?aeiia?aeii ca aeayeeie i?inoeie /eneaie q ? r, i?aea?oia F(Q oe?eeii oaeoi?eciaaia, i?aea?oia F(R aai oe?eeii oaeoi?eciaaia, aai p2-1 ae?eeoueny ia r ? aeey aoaeue-yei? ii?iaeueii? i?aea?oie F1 a?oie F(Q ?nio? ii?iaeueia i?aea?oia F2 a?oie F(R oaea, ui F=F1(F2, i?aea?oia Q(R iaaaaeaaa oe?eeii oaeoi?eciaaia; G=F((Q((a()(R1), aea Q- neeianueea q-i?aea?oia ?c G, R=(a((R1 - neeianueea r-i?aea?oia a?oie G, Q(R - iaaaaeaaa oe?eeii oaeoi?eciaaia a?oia, i?aea?oia F ?iceeaaea?oueny a i?yiee aeiaooie F=F1(…(Fk i?i?iaeueieo ii?iaeueieo i?aea?oi a?oie F((Q((a() c k=|R1|, ye? i?aea?oia R1 ia?anoaaey? i?ae niaith (ca aeiiiiiaith ni?yaeaiiy), eiaeia i?aea?oia a?oie Q((a( aai oeaio?ae?co? Fi, aai ae?? iaca?aeii ia Fi ?=1, …,k. A eai? 4.3.11 aeacaii eieaeueio aoaeiao ne?i/aiieo o?enooiaiaai ?ica’ycieo UH-a?oi, ii?yaeie yeeo ae?eeoueny ia iaio i?ae ia o?e ??cieo i?inoeo /enea, ? oi/a a iaeia c neeianueeeo i?aea?oi yeeo iaaaaeaaa. Ine?eueee ?? iiaia oi?ioethaaiy aeineoue a?ii?caeea ? i?noeoue iineeaiiy ia aeaye? oeie a?oi, ye? ?ai?oa aea/aa I.I.No/eia, oi aiii ooo ia iaaiaeeoueny.. Ianooiia oaa?aeaeaiiy iieaco?, ui iaiaaeaiiy a oai?ai? 4.3.12 ia nooi?iue ?ica’yciino? ia ? caiaaeoi iaoyaeeeaei. Oaa?aeaeaiiy 4.3.14. sseui ii?yaeie ne?i/aiii? UH-a?oie ae?eeoueny ia iaioa i?ae ia /ioe?e ??cieo i?inoeo /enea, oi ?? nooi?iue ?ica’yciino? ia ia?aaeuo? o?ueio. Inoaii?e i’yoee ?icae?e aeena?oaoe?eii? ?iaioe i?ena’y/aee aea/aiith aeaaa? E? oa a?oi (aano?aeoieo) c aeayeeie nenoaiaie ?ica’ycieo i?aeaeaaa? aai i?aea?oi. Oi/a a ?aeaeiiio a?aeiioaii? oe? iiiyooy o?nii iia’ycai? i?ae niaith, iaoiaee aea/aiiy aeaaa? E? c oeie /e ?ioeie nenoaiaie ?ica’ycieo i?aeaeaaa? ?noioii a?ae??ciythoueny a?ae iaoiae?a, ui canoiniaothoueny i?e aeine?aeaeaii? a?oi c aiaeia?/ieie aeanoeainoyie. Ia?oee ia?aa?ao iyoiai ?icae?eo i?noeoue iniiai? aeanoeaino? iaeaea ?ica’ycieo aeaaa? E?, ye? ? aiaeiaaie a?aeiia?aeieo aeanoeainoae iaeaea ?ica’ycieo a?oi (iaaaaea?ii, ui aeaaa?a E? iaae iieai iaceaa?oueny a aeena?oaoe?ei?e ?iaio? iaeaea ?ica’yciith, yeui aiia i?noeoue ?ica’yciee ?aeaae ne?i/aiii? eiaei??iino?). C?icoi?ei, ui aea/aoe eean iaeaea ?ica’ycieo aeaaa? E?, yeee i?noeoue an? ne?i/aiiiaei??i? ? an? ?ica’yci? aeaaa?e E?, iiaeia c oi/i?noth aei aeanoeainoae ne?i/aiiiaei??ieo ? ?ica’ycieo aeaaa? E?. Aaaeeeaith aeanoea?noth ne?i/aiiiaei??ieo aeaaa? E? ioeueiai? oa?aeoa?enoeee ? oa?aeoa?enoe/i?noue ?? (?ica’yciiai) ?aaeeeaeo). Oe??? aeanoeaino? aaea ia iathoue aeaaa?e E? iaae iieyie iiceoeaii? oa?aeoa?enoeee (a?aeiia?aeiee i?eeeaae i?noeoueny a eiec? , noi?. 88). I?e ia?aoiae? aei iaeaea ?ica’ycieo aeaaa? E? oey a?aei?ii?noue a iiaaae?ioe? aeaaa? ??cii? oa?aeoa?enoeee aaea cieea?, ye iieaco? ianooiia oai?aia, ui ? iaeiei ?c iniiaieo ?acoeueoao?a oeueiai ia?aa?aoo: Oai?aia 5.1.7. sseui L - iane?i/aiiiaei??ia iaeaea ?ica’ycia aeaaa?a E?, oi L i?noeoue oa?aeoa?enoe/iee ?ica’yciee ?aeaae ne?i/aiii? eiaei??iino?. I?e aeiaaaeaii? oe??? oai?aie iniiaiiei aoei ianooiia oaa?aeaeaiiy, aeiaaaeaiiy yeiai io?eiaia c aeei?enoaiiyi aeayeeo o?eueo?aoe?e iane?i/aiiiaei??ieo aeaaa? E? ca i?aeaeaaa?aie ne?i/aiii? eiaei??iino?: Eaia 5.1.6. Iaoae L - aeaaa?a E?, ? - iaeaea ?ica’yciee ?aeaae ?c L. Oiae? aeaaa?a L i?noeoue ?ica’yciee ?aeaae ?0 oaeee, ui ?0(? ? dimI/I0<(. Iieacaii, ui aiaeia oai?aie 5.1.7 aaea ia ia? i?noey aeey iaeaea eieaeueii i?eueiioaioieo aeaaa? E? iaae iieyie aeiaeaoiuei? oa?aeoa?enoeee, a naia, iieacaii, ui ?niothoue iane?i/aiiiaei??i? oa?aeoa?enoe/ii i?ino? aeaaa?e E?, ui i?noyoue eieaeueii i?eueiioaioiee ?aeaae ne?i/aiii? eiaei??iino?. Anoaiiaeaii oaeiae ianooii? e?eoa??? iaeaea ?ica’yciino? aeaaa?e E?: Oai?aia 5.1.10. sseui aeaaa?a E? L i?noeoue iaeaea ?ica’yciee ?aeaae I ? oaeoi?-aeaaa?a L/I iaeaea ?ica’ycia, oi aeaaa?a L iaeaea ?ica’ycia. Oai?aia 5.1.11. Iaoae L - aeaaa?a E?, yea i?noeoue ?ica’ycio i?aeaeaaa?o ne?i/aiii? eiaei??iino?. Oiae? L i?noeoue ?ica’yciee ?aeaae ne?i/aiii? eiaei??iino? (oiaoi, L iaeaea ?ica’ycia). A?aecia/eii, ui a oai?ai? 5.1.11 ?ica’ycia i?aeaeaaa?a ne?i/aiii? eiaei??iino? iiaea ia i?noeoe iaioeaaeo ?aeaae?a aeaaa?e L (oiaoi aiaeia oai?aie Ioaiea?a c oai??? a?oi ia ia? i?noey aeey iaeaea ?ica’ycieo aeaaa? E?). Oaa?aeaeaiiy oai?aie 5.1.11 iiaeia iineeeoe aeey aeaaa? E?, ye? i?noyoue aaaeaao i?aeaeaaa?o ne?i/aiii? eiaei??iino?, a naia: yeui aeaaa?a E? L i?noeoue aaaeaao i?aeaeaaa?o ne?i/aiii? eiaei??iino?, oi aiia i?noeoue aaaeaaee ?aeaae ne?i/aiii? eiaei??iino? (oiaoi L iaeaea aaaeaaa). A inoaiiueiio ia?aa?ao? aeena?oaoe?eii? ?iaioe aea/athoueny aeanoeaino? ia??iaee/ieo eieaeueii ?ica’ycieo a?oi a caeaaeiino? a?ae aeanoeainoae ?o neeianueeeo (-i?aea?oi (i?ae neeianueeith (-i?aea?oiith a?oie G ie ?icoi??ii iaeneiaeueio (-i?aea?oio a?oie G). ?icaeyaeathoueny neeianuee? (-i?aea?oie aeey ne?i/aiii? iiiaeeie (. Iniiaiei ?acoeueoaoii oeueiai ia?aa?aoo ? ianooiia oai?aia: Oai?aia 5.2.17. A ia??iaee/i?e eieaeueii ?ica’yci?e a?oi? G an? neeianuee? (-i?aea?oie iathoue ne?i/aii? aeniiiaioe oiae? ? o?eueee oiae?, eiee aiia ia? ne?i/aiiee ?yae oa?aeoa?enoe/ieo i?aea?oi, eiaeiee oaeoi? yeiai ? aai ?-a?oiith aeey ?(( ne?i/aiii? aeniiiaioe, aai (’-a?oiith. C oe??? oai?aie io?eia?ii oe?eaaee iane?aeie, yeee ia? naiino?eia cia/aiiy. Oai?aia 5.2.19. Eieaeueii ?ica’ycia a?oia ne?i/aiii? aeniiiaioe ia? ne?i/aiiee ?yae oa?aeoa?enoe/ieo i?aea?oi c i?eia?ieie oaeoi?aie. Ua iaeei ?acoeueoao oeueiai ia?aa?aoo a?aeiineoueny aei ?nioaaiiy a eieaeueii ne?i/aiieo eieaeueii (-?ica’ycieo a?oiao ?iaa??aioieo ?yae?a c (’-oaeoi?aie ? eieaeueii i?eueiioaioieie oaeoi?aie: Oai?aia 5.2.33. Iaoae G - eieaeueii ne?i/aiia eieaeueii (-?ica’ycia a?oia c ?icaycieie (-i?aea?oiaie ? n - iaeneioi ?o nooiai?a ?ica’yciino?. Oiae? yeui an? 2-i?aea?oie a?oie G aaaeaa?, oi G ia? ne?i/aiiee ?iaa??aioiee ?yae c (’-oaeoi?aie ? ?ica’ycieie nooiaiy (n eieaeueii i?eueiioaioieie (-oaeoi?aie, /enei inoaii?o a yeiio ia a?eueoa i?ae n. AENIIAEE A aeena?oaoe?ei?e ?iaio? aea/athoueny aeaaa?e E? (ia?aaaaeii iane?i/aiiiaei??i?), ye? ?iceeaaeathoueny a noio aeaio nai?o i?aeaeaaa?, ui ? i?eueiioaioieie aai aeecueeeie aei i?eueiioaioieo, a oaeiae anioe?aoeai? aeaaa?e, ui ? noiaie aeaio nai?o i?aeaeaaa?, ye? caaeiaieueiythoue oiiaai, aeecueeei aei eiiooaoeaiino? /e i?eueiioaioiino?. Iaeiei ?c iniiaieo ?acoeueoao?a aeena?oaoe?? ? iiaoaeiaa i?eeeaaeo ia?ica’ycii? ne?i/aiiiaei??ii? aeaaa?e E?, yea noiith aeaio nai?o i?eueiioaioieo i?aeaeaaa? ? oeei naiei ?ica’ycaiiiy (iaaaoeaii) a?aeiii? i?iaeaie I.Eaaaey i?i ?icayci?noue noie aeaio i?eueiioaioieo aeaaa? E?, iinoaaeaii? ia ii/aoeo 60-?ie?a. Aeiaaaeaii ?ica’yci?noue aeaaa?e E? (ia iaia’yceiai ne?i/aiiiaei??ii?) iaae aeia?eueiei iieai oa?aeoa?enoeee (2, yea ? noiith aaaeaai? oa i?eueiioaioii? i?aeaeaaa?. Aeine?aeaeaii iane?i/aiiiaei??i? aeaaa?e E? iaae aeia?eueiei iieai, ye? ? noiith aeaio i?aeaeaaa? c? ne?i/aiiiaei??ieie eiiooaioaie. Cie?aia aeiaaaeaii iaeaea ?ica’yci?noue oaeeo aeaaa?, aea/aii aeaye? aeanoeaino? FC-aeaaa? E?. ?icaeyiooi aeai?noo caaea/o i?i noio aeaio iaeaea aaaeaaeo aeaaa? E? (oiaoi oaeeo aeaaa?, ye? i?noyoue aaaeaaee ?aeaae ne?i/aiii? eiaei??iino?), aeiaaaeaii iaeaea ?ica’yci?noue oaeeo aeaaa? E? i?e oiia?, ui iaeei c aeiaeaie?a i?noeoue aaaeaaee ?aeaae a nai?io oeaio??. sse iane?aeie ca?aene io?eiaii, ui aeaaa?a E?, yea ? noiith aaaeaai? oa iaeaea aaaeaai? i?aeaeaaa? iaeaea ?ica’ycia. E??i cae/aeieo noi i?eueiioaioieo aeaaa? E? a aeena?oaoe?ei?e ?iaio? ?icaeyiooi ? iio??ei? noie, aeey ieo io?eiaii iien o aeiaaeeo, eiee an? aeiaeaiee ne?i/aiiiaei??i? aaaeaa? ? iaeei c ieo ? ?aeaaeii. Aea/aii anioe?aoeai? aeaaa?e iaae aeia?eueiei iieai, ye? ?iceeaaeathoueny a noio aeaio nai?o iaeaea eiiooaoeaieo i?aeaeaaa?, cie?aia aeiaaaeaii, ui oaea aeaaa?a i?noeoue i?eueiioaioiee ?aeaae, oaeoi?-aeaaa?a ca yeei iaeaea aaaeaaa. Oeae ?acoeueoao ? ocaaaeueiaiiyi ia anioe?aoeai? aeaaa?e ?acoeueoao?a, io?eiaieo TH.A.Aaooo??iei, A.Aeaeaia?oii, I.Eaaaeai aeey noi eiiooaoeaieo e?eaoeue. Io?eiaii oaeiae ocaaaeueiaiiy oai?aie I.Eaaaey i?i i?eueiioaioi?noue anioe?aoeaiiai e?eueoey, yea ? noiith aeaio nai?o i?eueiioaioieo i?aee?eaoeue, a naia aeiaaaeaii, ui anioe?aoeaia aeaaa?a iaae aeia?eueiei iieai, yea ? noiith aeaio nai?o iaeaea i?eueiioaioieo i?aeaeaaa? naia ? iaeaea i?eueiioaioiith. Aeiaaaeaii e??ao ?ica’yci?noue anioe?aoeaiiai e?eueoey, ui ?iceeaaea?oueny a noio eiiooaoeaiiai oa e??ai i?eueiioaioiiai i?aee?eaoeue. Io?eiaii ioe?ieo aeey nooiaiy e??ai? ?ica’yciino? oaeiai e?eueoey a caeaaeiino? a?ae eeano e??ai? i?eueiioaioiino? e??ai i?eueiioaioiiai aeiaeaieo. ?icaeyiooi ua iaeio caaea/o oaeoi?ecaoe?eiiai oa?aeoa?o, a naia i?i aoaeiao aeaaa? E?, a yeeo aeiiiaithaai? an? i?aeaeaaa?e aai o?eueee iaeiiaei??i? i?aeaeaaa?e (aeey ne?i/aiiiaei??ieo aeaaa? E? oe? oiiae ni?aiaaeathoue). Oey caaea/a, yea iioiaeeoue c oai??? a?oi, aeyaeeanue oaeiae o?nii iia’ycaiith c ae??th a?oi E? ia aeaaeeeo iaeiiaei??ieo iiiaiaeaeao (aai c iienii iaeii??aeieo i?inoi??a). A oeueiio oi?ioethaaii? aiia ?icaeyaeaeany aaaaoueia aaoi?aie iaae iieai oa?aeoa?enoeee 0, a aeena?oaoe?ei?e ?iaio? io?eiaii iien aeaaa? E? c aeiiiaithaaieie i?aeaeaaa?aie iaae aeineiiaeeie iieyie iiceoeaii? oa?aeoa?enoeee. I?e oeueiio aeyaeeeny i?eioeeiiaa ??cieoey i?ae oaeeie aeaaa?aie E? oa?aeoa?enoeee (2 ? oa?aeoa?enoeee ?=2. Aeey aeine?aeaeaiiy ye noi aeaaa? E?, oae ? aea/aiiy aeaaa? E? c nenoaiaie aeiiiaithaaieo i?aeaeaaa? aeyaeeinue iaiao?aeiei aeine?aeeoe iane?i/aiiiaei??i? aeaaa?e E? c oeie /e ?ioeie nenoaiaie ?ica’ycieo i?aeaeaaa?. Aeiaaaeaii, ui aeaaa?a E? iaae aeia?eueiei iieai, yea i?noeoue ?ica’ycio i?aeaeaaa?o ne?i/aiii? eiaei??iino? aoaea iaeaea ?ica’yciith, oiaoi aoaea i?noeoe ?ica’yciee ?aeaae ne?i/aiii? eiaei??iino?, iieacaii oaeiae, ui iaeaea ?ica’ycia aeaaa?a E? i?noeoue oa?aeoa?enoe/iee ?ica’yciee ?aeaae ne?i/aiii? eiaeii?iino?. Aea/aii iane?i/aii? a?oie, a yeeo aeiiiaithaai? iai?eia?i? iaoeeee?/i? i?aea?oie, a oaeiae i?e aeayeeo oiiaao ne?i/aii? a?oie, a yeeo aeiiiaithaai? an? iaaenee?anuee? i?aea?oie. Io?eiaii oa?aeoa?ecaoe?th ia??iaee/ieo eieaeueii ?ica’ycieo a?oi c? ne?i/aiiiaeniiiaioieie neeianueeeie (-i?aea?oiaie. ?IAIOE AAOI?A CA OAIITH AeENA?OAOe?? Iao?aa/oe A.I. Aaneiia/iua a?oiiu n aeiiieiyaiuie iaoeeeee/aneeie iai?eia?iuie iiaea?oiiaie // Na. A?oiiu e nenoaiu eo iiaea?oii. - Eeaa: Ei-o iaoaiaoeee AI ONN?. - 1983. - N.73-79. Iao?aa/oe A.I. Eiia/iua a?oiiu, a eioi?uo aeiiieiyaiu ana iaaeneeianeea iiaea?oiiu // Na. No?iaiea a?oii e naienoaa eo iiaea?oii. - Eeaa: Ei-o iaoaiaoeee AI ONN?. - 1986. - N.74-82. Iao?aa/oe A.I. Aeaaa?u Ee, ?aceiaeeiua a noiio aaaeaaie e ieeueiioaioiie iiaeaeaaa? // Oe?. iaoai. aeo?i.- 1988. - 40, ? 3. - N.385-388. Iao?aa/oe A.I. I ?ac?aoeiinoe aeaaa?u Ee, ?aceiaeeiie a noiio aaaeaaie e ieeueiioaioiie iiaeaeaaa? // Oe?. iaoai. aeo?i.- 1991. - 43, ? 7-8. - N.986-991. Iao?aa/oe A.I. I noiia aeaoo aeaaa? Ee n eiia/iiia?iuie eiiiooaioaie// Oe?. iaoai. aeo?i.- 1995. - 47, ? 8. - N.1089-1096. Iao?aa/oe A.I. I aaneiia/iiia?iuo aeaaa?ao Ee n ?ac?aoeiuie eaeaaeaie eiia/iie ei?acia?iinoe// Na. Aeaaa?ae/aneea enneaaeiaaiey. - Eeaa: Ei-o iaoaiaoeee IAI Oe?aeiu. - 1996. - N.158-167. Iao?aa/oe A.I. I?i noio eiiooaoeaiiai oa e??ai i?eueiioaioiiai anioe?aoeaieo e?eaoeue// A?niee Ee?anueeiai oi?aa?neoaoo, “Iaoaiaoeea ? iaoai?ea”. - 1998. - aei.1. - N.33-35. Iao?aa/oe A.I. I noiia ii/oe aaaeaaie aeaaa?u Ee e aeaaa?u Ee, eiia/iiia?iie iaae naiei oeaio?ii// Oe?. iaoai. aeo?i. - 1999. - 51, ? 5. - N.636-644. Iao?aa/oe A.I. Aeaaa?e E? c aeiiiaithaaieie iaeiiaei??ieie i?aeaeaaa?aie// A?niee Ee?anueeiai oi?aa?neoaoo, “Iaoaiaoeea ? iaoai?ea”. - 1999. - aei.3. - N.32-37. Iao?aa/oe A.I. I eeaaie ?ac?aoeiinoe noiiu eiiiooaoeaiiai e eeaai ieeueiioaioiiai annioeeaoeaiuo eieaoe // A?niee Ee?anueeiai oi-oo, Na?. O?c.-iao.- 1999. - ? 1. - N.78-81. -iiaea?oiiaie// Oe?. iaoai. aeo?i. - 1987. - 39, ? 6. - N.761-767. -?ac?aoeiuo a?oii // Na. Aaneiia/iua a?oiiu e i?eiueathuea aeaaa?ae/aneea no?oeoo?u. - Eeaa: Ei-o iaoaiaoeee IAI Oe?aeiu. - 1993. - N.393-404. Petravchuk A.P. On triple sums of abelian Lie algebras // Iaoaiaoe/i? nooae??. - 1997. - 8, ? 1. - ?.11-14. Petravchuk A.P. On associative algebras which are sum of two almost commutative subalgebras // Publicationes Mathematicae (Debrecen) - 1998. - 53, ? 1-2. - P.191-206. Petravchuk A.P. Locally finite Lie algebras with complemented subalgebras // Iaoaiaoe/i? nooae??. - 1999. - 11, ? 2. - ?.135-140. Iao?aa/oe A.I. Eiia/iiia?iua aeaaa?u Ee, ?aceiaeeiua a noiio aaaeaaie e ieeueiioaioiie iiaeaeaaa? // XIX Ananithc. Aeaaa?. Eiio. (Oac. Aeiee.) - Eueaia. - 1987. - *.1. - N.216. Iao?aa/oe A.I. Eiia/iiia?iua aeaaa?u Ee, ?aceiaeeiua a noiio aeaoo ieeueiioaioiuo iiaeaeaaa? // Iaaeaeoia?iaeiay aeaaa?. Eiio. Iaiyoe A.E.Iaeueoeaaa (Oac. Aeiee.) - Iiaineae?ne, 1989. - N.103. Iao?aa/oe A.I. I noiia aaaeaaie e ieeueiioaioiie (eeanna 2) aeaaa? Ee // Iaaeaeoia?iaeiay aeaaa?. Eiio. Iaiyoe A.E.Oe?oiaa (Oac. Aeiee.) - Aa?iaoe, 1991, - N.91. Iao?aa/oe A.I. Aeaaa?e E? c ?icaycieie ?aeaaeaie ne?i/aiii? eiaei??iino?// Anaoe?a?inueea iaoaiaoe/ia eiioa?aioe?y iai’yo? I.Eacei??nueeiai (Oac. Aeiiia.) - Euea?a, 1995. - *.1. - N.40. Iao?aa/oe A.I. I?i noio aaaeaai? oa iaeaea aaaeaai? aeaaa? E?// Anaoe?a?inueea iaoaiaoe/ia eiioa?aioe?y iai’yo? I.Eacei??nueeiai (Oac. Aeiiia.) - Euea?a, 1995. - *.1. - N.40. Iao?aa/oe A.I. I?i iio??ei? noie aaaeaaeo aeaaa? E? // 5-oa I?aeia?iaeia iaoaiaoe/ia eiioa?aioe?y iai’yo? aeaae.I.I.E?aa/oea (Oac. Aeiiia.). - Ee?a, 1996. - N.329. Iao?aa/oe A.I. I noiia eiiiooaoeaiuo e eeaai ieeueiioaioiuo annioeeaoeaiuo eieaoe // Iaaeaeoia?iaeiay eiioa?aioeey ii oai?ee a?oii, iinayuaiiay iaiyoe N.I.*a?ieeiaa. (Oac. Aeiee.). - Ia?iue. - 1997. - N.52. *a?ieeia I.N., Iao?aa/oe A.I. Iaeioi?ua e?eoa?ee ?ac?aoeiinoe e ?aaeeeaeueiinoe ia?eiaee/aneeo eieaeueii ?ac?aoeiuo a?oii // O Ananithc. Neiiiceoi ii oai?ee a?oii (Oac. Aeiee.). - Aiiaeue, 1986. - N.255. *a?i?eia I.N., Iao?aa/oe A.I. Aei oai?aie O.Oieea ? A.O?aiaia // I?aeia?iaeia iaoaiaoe/ia eiio. Iaiyo? aeaae. I.I.E?aa/oea (Oac. Aeiiia.) - Ee?a, 1993. - N.234. Petravchuk A.P. On sum of two associative commutative algebras // International algebraic Conference dedicated to the memory of D.K.Faddeev, (Absracts). - St. Petarsburg, 1997. - ?.95-96. Petravchuk A.P. On sum of two Lie algebras which are near to abelian // Intranational algebraic conference dedicated to the memory of L.M.Gluskin, (Abstracts). - Slovyansk, 1997. - P.109. Petravchuk A.P. On sum of abelian and nilpotent Lie algebras // The Fifth International Conference “Groups and Group Rings” (Abstracts). - 1997. - P.19. Iao?aa/oe A.I.Oaeoi?ecaoe?y ? aeiiiaithaai?noue a aeaaa?ao E? ? anioe?aoeaieo aeaaa?ao.- ?oeiien. Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy aeieoi?a o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.06 - aeaaa?a ? oai??y /enae.- Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea, Ee?a, 1999. Aeena?oaoe?y i?ena’y/aia aeine?aeaeaiith noi i?eueiioaioieo aai aeecueeeo aei i?eueiioaioieo aeaaa? E?, a oaeiae aea/aiith noi anioe?aoeaieo aeaaa? ? e?eaoeue, ye? aeecuee? aei i?eueiioaioieo aai eiiooaoeaieo. Io?eiaii iaaaoeaiee ?ica’ycie a?aeiii? i?iaeaie I.Eaaaey i?i oa, /e aoaea ?ica’yciith aeaaa?a E?, yea ?iceeaaea?oueny a noio aeaio nai?o i?eueiioaioieo i?aeaeaaa?. Aeine?aeaeaii aeaaa?e E?, ye? ? noiaie aeaio i?aeaeaaa? c? ne?i/aiiiaei??ieie eiiooaioaie oa aea/aii noie iaeaea aaaeaaeo aeaaa? E?. Aea/aii aeaye? eeane aeaaa? E?, a yeeo aeiiiaithaai? an? iaeiiaei??i? i?aeaeaaa?e, anoaiiaeaii iniiai? aeanoeaino? aeaaa? E? c ?icaycieie i?aeaeaaa?aie ne?i/aiii? eiaei??iino?. Aeine?aeaeaii anioe?aoeai? aeaaa?e, ui ?iceeaaeathoueny a noio aeaio nai?o iaeaea eiiooaoeaieo i?aeaeaaa?. Iienaii eieaeueii nooi?i/ano? a?oie c aeiiiaithaaieie iai?eia?ieie iaoeeee?/ieie i?aea?oiaie, aeaye? eeane a?oi c aeiiiaithaaieie iaaenee?anueeeie i?aea?oiaie oa ioa?aeoa?eciaaii ia??iaee/i? eieaeueii ?ica’yci? a?oie c? ne?i/aiiiaeniiiaioieie neeianueeeie (-i?aea?oiaie. Eeth/ia? neiaa: noia aeaaa? E?, noia anioe?aoeaieo aeaaa?, i?eueiioaioia aeaaa?a E?, ?ica’yciee ?aeaae, nooi?iue ?ica’yciino? aeaaa?e E?, e??ai i?eueiioaioia anioe?aoeaia e?eueoea, aeiiiaithaaia i?aeaeaaa?a, eieaeueii ne?i/aiiiaei??ia aeaaa?a E?, eieaeueii ?ica’ycia ia??iaee/ia a?oia, aeiiiaithaaia i?aea?oia. Iao?aa/oe A.I.. Oaeoi?ecaoeey e aeiiieiyaiinoue a aeaaa?ao Ee e annioeeaoeaiuo aeaaa?ao. - ?oeiienue. Aeenna?oaoeey ia nieneaiea o/aiie noaiaie aeieoi?a oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.06 - aeaaa?a e oai?ey /enae.- Eeaaneee oieaa?neoao ei. O.Oaa/aiei, Eeaa, 1999. Aeenna?oaoeey iinayuaia enneaaeiaaieth noii ieeueiioaioiuo eee aeeceeo e ieeueiioaioiui aeaaa? Ee, a oaeaea eco/aieth noii annioeeaoeaiuo aeaaa? e eieaoe, eioi?ua aeecee e ieeueiioaioiui eee eiiiooaoeaiui. Iieo/aii io?eoeaoaeueiia ?aoaiea ecaanoiie i?iaeaiu I.Eaaaey i oii aoaeao ee ?ac?aoeiie aeaaa?a Ee, ?aceiaeeiay a noiio aeaoo naieo ieeueiioaioiuo iiaeaeaaa?. Enneaaeiaaiu aeaaa?u Ee, yaeythueany noiiaie aeaoo iiaeaeaaa? n eiia/iiia?iuie eiiiooaioaie e eco/aiu noiiu ii/oe aaaeaauo aeaaa? Ee. Eco/aiu iaeioi?ua eeannu aeaaa? Ee, a eioi?uo aeiiieiyaiu ana iaeiiia?iua iiaeaeaaa?u, onoaiiaeaiu iniiaiua naienoaa aeaaa? Ee n ?ac?aoeiuie iiaeaeaaa?aie eiia/iie ei?acia?iinoe. Enneaaeiaaiu annioeeaoeaiua aeaaa?u, ?aceiaeeiua a noiio aeaoo naieo ii/oe eiiiooaoeaiuo iiaeaeaaa?. Iienaiu eieaeueii nooiai/aoua a?oiiu n aeiiieiyaiuie iai?eia?iuie iaoeeeee/aneeie iiaea?oiiaie, iaeioi?ua eeannu a?oii n aeiiieiyaiuie iaaeneeianeeie iiaea?oiiaie e ioa?aeoa?eciaaiu ia?eiaee/aneea eieaeueii ?ac?aoeiua a?oiiu n eiia/iiyeniiiaioiuie neeianeeie (-iiaea?oiiaie. Eeth/aaua neiaa: noiia aeaaa? Ee, noiia annioeeaoeaiuo aeaaa?, ieeueiioaioiay aeaaa?a Ee, ?ac?aoeiue eaeaae, nooiaiue ?ac?aoeiinoe aeaaa?u Ee, eeaai ieeueiioaioiia annioeeaoeaiia eieueoei, aeiiieiyaiay iiaeaeaaa?a, eieaeueii eiia/iiia?iay aeaaa?a Ee, eieaeueii ?ac?aoeiay ia?eiaee/aneay a?oiia, aeiiieiyaiay iiaea?oiia. Petravchuk A.P.. Factorization and complementation in Lie algebras and associative algebras. Manuscript. Thesis of the dissertation for obtaining of the degree of doctor of sciences in physics and mathematics, speciality 01.01.06 - algebra and number theory. Kyiv Taras Shevchenko University, Kyiv, 1999. The dissertation is devoted to investigating sums of nilpotent or near to nilpotent Lie algebras and sums of two associative algebras which are near to commutative or nilpotent. The negative solution of the known problem of O.Kegel about solubility of sum of two nilpotent Lie algebras is obtained. The solubility of a Lie algebra over every field of characteristic (2 which is a sum of an abelian subalgebra and a nilpotent subalgebra is proved. It is also proved that every Lie algebra over an arbitrary field which can be decomposed in a sum of two subalgebras with finite dimensional commutants (derived algebras) is almost soluble (i.e. it contains a soluble ideal of finite codimension). Every Lie algebra with an abelian ideal of finite codimension (almost abelian Lie algebra) is also near to abelian, and for sums of such algebras the following assertion is obtained: if a Lie algebra L over an arbitrary field is a sum of an almost abelian subalgebra A and a subalgebra B which is finite over its centre, then L is almost soluble. In particular, every sum of an abelian and an almost abelian Lie algebras is almost soluble. It is shown that every associative algebra over an arbitrary field which is a sum of two almost commutative subalgebras contains a nilpotent ideal with almost commutative quotient algebra (particularly, this sum is a PI-algebra). The Lie solubility of an associative ring which is a sum of a commutative subring and a Lie nilpotent subring is obtained. Some estimations of the derived length of these sums that depend on Lie nilpotency class of the Lie nilpotent summand are pointed out. Finite dimensional Lie algebras with complemented subalgebras over perfect fields are described. Locally finite dimensional Lie algebras with complemented one-dimensional subalgebras over perfect fields are characterized. The main properties of almost soluble Lie algebras are stated: every almost soluble Lie algebra has a characteristic soluble ideal of finite codimension; every extension of an almost soluble Lie algebra by other almost soluble Lie algebra is also almost soluble; every Lie algebra with an soluble subalgebra of finite codimension is almost soluble. Infinite groups with complemented non-primar non-cyclic subgroups are described. Finite groups with complemented over-Sylow subgroups (i.e.subgroups which contain a non-identity Sylow-subgroup) are characterized. It is proved that in a periodic locally soluble group G all (-subgroups (( is a finite set of primes) are of finite exponent iff G has a finite series of characteristic subgroups, whose all factors are either p-groups for p(( of finite exponent, or (’-groups. Key words: sum of Lie algebras, sum of associative algebras, nilpotent Lie algebra, soluble ideal, derived length of a Lie algebra, Lie nilpotent associative ring, complemented subalgebra, locally finite-dimensional Lie algebra, locally soluble periodic group, complemented subgroup. Kegel O.H. Zur Nilpotenz gewisser assoziativer Ringe // Math. Ann. - 1963. 149. - S.258-260 Eio?ianeay oao?aaeue (Ia?aoaiiua aii?inu oai?ee a?oii), 11 ecae. - Iiaineae?ne: Ei-o iaoaiaoeee NI AI NNN?, 1990. - 126n. Goto M. Note on a characterization of solvable Lie algebras // J.Sci. Hiroshima Univ. Ser. A1. -1962. - 26, N1. P.1.-2 Amberg B., Franciosi S., de Giovanni F. Products of groups. - Oxford, Clarendon Press, 1992. *a?ieeia I.N. I i?iecaaaeaiee ii/oe aaaeaauo a?oii // Oe?. iao. aeo?i. - 1981. - 33, N1. - N.136-138. Nunae ss.I. I i?iecaaaeaieyo ii/oe aaaeaauo a?oii// Na. Enneaaeiaaiey a?oii n ia?aie/aieyie aeey iiaea?oii. - Eeaa: Ei-o iaoaiaoeee AI ONN?. - 1988. - N.81-86.. Nunae ss.I. I?iecaaaeaiey aaneiia/iuo a?oii . - Eeaa, 1982. - 36n. - (I?ai?eio/ AI ONN?, Ei-o iaoaiaoeee; N82.53). Oiaeueeei A.E. Nieaoaiea aeaaa? Ee e eo i?eiaiaiea a oai?ee a?oii // O?oaeu Iine. Iaoai. ia-aa. - 1973. - 29. -N.247-260. Herstein I.N., Small L.W. Nil rings satisfying certain chain conditions// Canad. J.Math. - 1964. - 16. - P.771-776. Ferrero M., Puczylowski E.R. On rings which are sums of two subrings // Arch. Math.. -1989. - 53. - P.4-10. Aaeaea? E.E., Ieoaeaa A.A. Iaiauaiiua iieeiiieaeueiua oiaeaeanoaa e eieueoea, yaeythueany noiiaie aeaoo iiaeeieaoe // Aeaaa?a e eiaeea. - 1995. - 34a ?1. - N.3-11. Aiae?oiaeeaae/ A.A. , ?yaooei TH.I. ?aaeeeaeu aeaaa? e no?oeoo?iay oai?ey. - I.:Iaoea. - 1979. - 495n. *a?ieeiaa I.A. A?oiiu n aeiiieiyaiuie iiaea?oiiaie // Iaoai. na. - 1956. - 3. - N.273-292. Aeaeoiaaeeeueaeaaa A.N. I?inoua aeaaa?u Ee n iiaeaeaaa?ie ei?acia?iinoe 1 // OII. - 1985. - 40, N1.- N.193-194. Hofmann K.H. Hyperplane subalgebras of real algebras// Geometriae Dedicata. - 1990. - 36, N2-3. - P.207-224. *a?ieeia I.N., Iao?aa/oe A.I. Ia iaeiii oneiaee aeiiieiyaiinoe// Na. No?iaiea a?oii e naienoaa eo iiaea?oii. - Eeaa: Ei-o iaoaiaoeee AI ONN?. - 1978. - N.131-146. Aeaeaeianii I. Aeaaa?u Ee. - I.:Iaoea, 1964 PAGE PAGE 3 PAGE aaaaaaaaaaaaaaaaaaaaaaaa PAGE 3

Похожие записи