Напівгрупи та майжекільця перетворень: Автореф. дис… д-ра фіз.-мат. наук / В.М. Усенко, Київ. ун-т ім. Т.Шевченка. — К., 1999. — 32 с. — укp.

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

Ia i?aaao ?oeiieno

Onaiei A?oae?e Ieoaeeiae/

OAeE 512.534+512.558

IAI?AA?OIE OA IAEAEAE?EUeOess IA?AOAI?AIUe

01.01.06. Aeaaa?a oa oai??y /enae

Aaoi?aoa?ao

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

aeieoi?a o?ceei-iaoaiaoe/ieo iaoe

Ee?a-2000

Aeena?oaoe??th ? ?oeiien.

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

I?i?noa?noaa ina?oe Oe?a?ie.

Iaoeiaee eiinoeueoaio aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani? Ee?e/aiei Aieiaeeie? Aaneeueiae/, Ee?anueeee iaoe?iiaeueiee

oi?aa?neoao ?iai? O.Oaa/aiea,

caa?aeoaa/ eaoaae?ith aaiiao???.

Io?oe?ei? iiiiaioe aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani? A?eai?/oe ?inoeneaa ?aaiiae/,

i?ia?aeiee iaoeiaee ni?a?ia?oiee

Iaoaiaoe/iiai ?inoeoooo ?i. A.A.No?eeiaa

?in?enueei? Aeaaeai?? Iaoe;

aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani? Iii?cianueeee ?ineo Nieiiiiiae/,

?in?enueeee aea?aeaaiee a?ae?iiaoa?ieia?/iee

oi?aa?neoao, i. Naieo-Iaoa?ao?a

aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani? I?ioania ?ai? Aieiaeeie?iae/,

i?ioani? eaoaae?e aeine?aeaeaiiy iia?aoe?e,

Ee?anueeee Oi?aa?neoao ?iai? O.Oaa/aiea,

I?ia?aeia onoaiiaa Euea?anueeee aea?aeaaiee Oi?aa?neoao

?i. ?aaia O?aiea, i.Euea?a

Caoeno a?aeaoaeaoueny «_14_» __ethoiai__ _2000_?. i __14_ aiaeei? ia
can?aeaii?

niaoe?ae?ciaaii? a/aii? ?aaeeAe26.001.18 a Ee?anueeiio iaoe?iiaeueiiio

oi?aa?neoao? ?iai? Oa?ana Oaa/aiea ca aae?anith:252127 Ee?a-127,

i?iniaeo aeaaeai?ea Aeooeiaa 6,

iaoai?ei-iaoaiaoe/iee oaeoeueoao.

C aeena?oaoe??th iiaeia iciaeiieoenue a a?ae?ioaoe? oi?aa?neoaoo

(aoe. Aieiaeeie?nueea 62)

Aaoi?aoa?ao ?ic?neaiee «_27_» __a?oaeiy_ __1999_ ?ieo.

A/aiee nae?aoa?

niaoe?ae?ciaaii? ?aaee Iao?aa/oe A.I.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeueiinoue oaie. Iaeii?th c oa?aeoa?ieo iciae no/aniiai aoaio
?icaeoeo aeaaa?e ? iiaeaeuaiiy aeoeaiinoi aeineiaeaeaiue a iaeanoyo,
i?iiiaeieo aiae oai?ie eeane/ieo aeaaa?a?/ieo nenoai (a?oi, eieaoeue oa
iiaeoeia, anioeiaoeaieo aeaaa? oiui) c iaeiiai aieo aei caaaeueii?
oai?i? oiiaa?naeueieo aeaaa? — c iioiai. Oea yaeua i?iaiicoaaeinue
I.A.Eo?ioai ye oaea, ui aiaeiiaiaea? iniiaiei oaiaeaioeiyi ?icaeoeo
no/anii? caaaeueii? aeaaa?e (aeea., iai?eeeaae, anooi aei iiiia?aoi?
«Eo?io A.A. Iauay aeaaa?a (eaeoeee 1969-1970 o/.a.) // I.: Iaoea.-
1974.- 160n.).

Aei oaeeo i?iiiaeieo iaeanoae iaeaaeeoue i oai?iy iaeaeaeieaoeue, ui
nai?ie aeoieaie nyaa? ?iaio Aeieniia 1905?. (Dickson L. Definitions of
group and a field by independent postulates // Trans. Amer. Math. Soc.-
6.- p.198-204; Dickson L. On finite algebras // Nachr. Acad. Wiss.
Goettingen.- 1905.- p.358-393).

?icaeoie oai?i? iaeaeaeieaoeue iiaeiey?oueny ia o?e iniiaieo aoaie.
Ia?oee aoai (aei ii/aoeo 40-o ??.) — aeineiaeaeaiiy iniiaieo caaaeueieo
aeanoeainoae iaeaeaeieaoeue, iienaiiy aeayeeo eeania neii/aiieo
iaeaeaeieaoeue, canoinoaaiiy a oai?i? a?oi iiaenoaiiaie (Oeannaioaoc,
I?a, Oaonnueea, Oiooiia, Aaaeai, Aaaeaea?aa?i).

Ia ae?oaiio aoaii naiai ?icaeoeo (ii/aoie 40-o ?? — eiiaoeue 70-o ??.)
oai?iy iaeaeaeieaoeue canoiniao?oueny a caaea/ao eeaneoieaoei?
iaeiiieieo iaoaiaoe/ieo no?oeoo? (Menger K. Algebra of analysis // Notre
Dame Math. Lect.- 1944.- N3; Menger K. Tri-operational algebra // Report
of a Math. Colloq. Second Series, Issue 5-6, Notre Dame.- 1944.- p.3-10;
. Jordan P. Ueber polynomiale Fastringe // Acad. Wiss. Mainz. Math.-
Nat. Kl.- 1951.- p.337-340). Caeeaaeaii iniiae no?oeoo?ii? oai?i?
iaeaeaeieaoeue (Blackett D. W. Simple and semisimple near-rings // Proc.
Amer. Math. Soc.- 1953.- 4.- p.772-785). Aeyaeaii ca‘ycee c oai?i?th
a?oiiaeo iiiaiaeaeia (Neumann H. On varieties of groups and their
associated near-rings // Math. Z.- 1956.- 65.- pp.36-69). A oa?iiiao
oai?i? iaeaeaeieaoeue iieeaaeaii ii/aoie iaeiiooaoeaii? aiiieiai/ii?
aeaaa?e (O?ueieio, na?iy ?iaio 1959-1962 ??). A oeae ia?iiae aeyaeaii
i?inoioo iaeaeaeieaoeue aeayeeo eeania i, cie?aia, iiaieo iaeaeaeieaoeue
ia?aoai?aiue a?oi (=neiao?e/ieo iaeaeaeieaoeue ia a?oiao) oa
iieiiiiiaeueieo iaeaeaeieaoeue iaae iieyie ioeueiai? oa?aeoa?enoeee.
Io?eiaii aiaeiae oai?ai Aeaeaeianiia, Aaaeaea?aa?ia-A?oiia, aeayeeo
iioeo oai?ai no?oeoo?ii? oai?i? eieaoeue (Berman, Silverman, Betsch,
Mlitz, Scott oa ii.). Iieeaaeaii ii/aoie oai?i? ?aaeeeaeia
iaeaeaeieaoeue (Betsch, Meldrum, Holcombe, Pilz, Oswald, Angerer,
E.Eaa?ei oa ii.). C‘yaeeenue iaeyaee oa iiaenoieiai iiiia?aoi? c oai?i?
iaeaeaeieaoeue (Pilz G. Near-rings // North-Holland American Elsevier,
Amsterdam.- 1977.- 464p.).

O?aoie aoai ?icaeoeo oai?i? iaeaeaeieaoeue — no/aniee. Iiaeeaeth?oueny
eeaneoieaoeiy iaeaeaeieaoeue, /eneaiiioeie noathoue iaeaeaeieueoeaai
eiino?oeoei?, iioe?ththoueny canoinoaaiiy. Aeoeaii ?icaeaa?oueny oai?iy
?aaeeeaeia. Ii/aeiny aea/aiiy iao?e/ieo, a?oiiaeo oa iaiiaa?oiiaeo
iaeaeaeieaoeue (Mason, Ligh, Meldrum oa ii.), iioeo aeaaa?a?/ieo nenoai,
iia‘ycaieo c iaeaeaeieueoeyie, c‘yaeeenue aeayei ocaaaeueiaiiy (Andre,
Bachman, Williams, Ferrero, Cotti, Veldsman, Stefanescu, Velasco oa
ii.). Iaoiaee oai?i? iaeaeaeieaoeue ii/aee aiaeia?aaaoe nooo?ao ?ieue a
eeaneoieaoei? e?aoiio?aiceoeaieo a?oi iiaenoaiiaie (Kerby, Karzel,
Wefelscheid, Hille, Boehm, Bohnenstengel oa ii.).

Aei i?iaeaiaoeee oai?i? iaeaeaeieaoeue caa?oaeenue I.A.Eo?io, A.
I.Ieioeii oa ?o o/ii. Aeayeeie ieoaiiyie oai?i? iaeaeaeieaoeue oa ??
ca‘yceai c oai?i?th ioeueoeiia?aoi?ieo a?oi oiui i?enay/aii ?iaioe
N.A.Iieiia, TH.A.Eocueiiia, A.A.Ia?eia, E.Eaa?ei.

A Oe?a?ii iaeii?th c ia?oeo ?iaio, a yeeo ?icaeyaeathoueny
iaeaeaeieueoey, ? ?iaioa «Eaeoaeiei E.A., Nouaineee A.E. Aa?aaeueiua
ooieoeee ia a?oiiao // Oai?. e i?eee. aii?. aeeo. o?. e aeaaa?u. E.:
Ioeiaa aeoiea.- 1977.- n.105-110». Iiciioa c‘yaeeeny ?iaioe, iia‘ycaii c
aea/aiiyi a?oiiaeo aiaeia?aaeaiue oa iaeaeaeieaoeue ia?aoai?aiue
(Ee?e/aiei A.A., Onaiei A.I., Ei?oaaeca E.A., Iioaeeiaa I.I., ?yaooi
I.I.).

Eiinoaooth/e aenieee ?iaaiue ?icaeoeo aeineiaeaeaiue c oai?i?
iaeaeaeieaoeue neiae cacia/eoe, ui iaeii?th c oeaio?aeueieo i?iaeai
oei?? oai?i? caeeoa?oueny i?iaeaia eeaneoieaoei? iaeaeaeieaoeue
ia?aoai?aiue a?oi.

*anoeio o?oaeiiuia a oeie iaeanoi c iaeiiai aieo coiiaeaii iaiaaeaiinoth
eeane/ieo oaeoi?ecaoeieieo iaoiaeia, oa aiaenooiinoth aaeaeaaoieo
iaeaeaeieueoeaaeo eiino?oeoeie — c iioiai. Iaaeineoue oaaae i?eaeieyeiny
aeei?enoaiith ioeueoeieieaoeaieo aeanoeainoae iaeaeaeieaoeue
ia?aoai?aiue. Aeiaaee /an eeoaany iaaeei?enoaiei iioaioeiae aeayeeo
eiioeaioeie, ui aeieeee a oai?i? a?oi oa a oai?i? iaiiaa?oi, iiaeeeainoi
yeeo aeoiaeeee aeaeaei ca iaaei oeeo oai?ie. Oea noino?oueny ia?o ca ana
eiioeaioei?, ui iieyaaea a ?aaeicaoei? aiaeiioaiue iiae aeaaa?a?/ieie
nenoaiaie o aeaeyaei eaoaai?ie ia? oaeeo nenoai. C ii/aoeo 60-o ?ieia
oeth eiioeaioeith aoei aoieaii A. I.Ieioeiiei oa eiai oeieith a oai?i?
a?oiiaeo ia? c oe?iei ?icaaeoaeaieie iiaeaeueoeie canoinoaaiiyie a
oai?i? cia?aaeaiue a?oi a?oiaie aaoiii?oiciia aeaaa?a?/ieo nenoai. A
oai?i? iaiiaa?oi a oie aea ia?iiae eiioeaioeith nia?thaaiiy (ni?yaeaiiy)
aeaaa?a?/ieo nenoai aoei ?aaeiciaaii oai??th iaiiaa?oi aiaeiii?oiciia
iaoaiaoe/ieo no?oeoo? (a ?icoiiiii Ao?aaei), ii/aoie yei? aoei
iieeaaeaii E.I.Aeoneiiei na?i?th ?iaio i?i aecia/oaaiinoue aeayeeo
eeania iaoaiaoe/ieo no?oeoo? nai?ie iaiiaa?oiaie aiaeiii?oiciia. I?e
oeueiio a?oiaie aaoiii?oiciia oaei no?oeoo?e ye i?aaeei ia aecia/aeeny.
A iniiai iaoiaeia oai?i? iaiiaa?oi aiaeiii?oiciia aoei iieeaaeaii iaoiae
uieueieo ?icoe?aiue (E.I.Aeoneii, A.I.Oaei, E.I.Oaa?ii, I.Iao?i/).

Aeooaeueiinoue oaie oei?? aeena?oaoeieii? ?iaioe aecia/aii ?? iaoith:
?iciianthaeaeaiiy eiioeaioei? ni?yaeaiiy aeaaa?a?/ieo nenoai ia oai?ith
iaeaeaeieaoeue i canoinoaaiiy ?? aei i?iaeaie eeaneoieaoei?
iaeaeaeieaoeue ia?aoai?aiue. Iniiaieie caaea/aie i?e oeueiio ?

• iienaiiy no?oeoo?ieo aeanoeainoae iaeaeaeieaoeue ia?aoai?aiue oa ?o
ioeueoeieieaoeaieo iaiiaa?oi, yeeie aiie aecia/athoueny c oi/iinoth aei
iciii?oicio (i?iaeaia aano?aeoii? oa?aeoa?ecaoei?);

• iienaiiy oaeoi?ecaoeieieo aeanoeainoae iaeaeaeieaoeue ia?aoai?aiue
a?oi, ioeueoeieieaoeaieo iaiiaa?oi oaeeo iaeaeaeieaoeue, ui
aecia/athoueny oaeoi?ecaoeieieie aeanoeainoyie aiaeiiaiaeieo a?oi.

?iaioa aeeiio?oueny ca oaiaoeeith iaoeiaeo aeineiaeaeaiue eaoaae?e
aeaaa?e oa iaoaiaoe/ii? eiaiee Ee?anueeiai oiiaa?neoaoo iiaii Oa?ana
Oaa/aiea oa eaoaae?e aeaaa?e Neia‘yinueeiai aea?aeaaiiai iaaeaaiai/iiai
iinoeoooo i, cie?aia, o aiaeiiaiaeiinoi c i?ia?aiaie

• IAe? «Oai?iy aeaaa?a?/ieo nenoai oa ?o cia?aaeaiue i ?? canoinoaaiiy»
(aea?ae. ?a?no?. N 0197U003160) ca eiiieaeniith iaoeiaith i?ia?aiith
Ee?anueeiai Oiiaa?neoaoo iiaii Oa?ana Oaa/aiea «Iiaoaeiaa oa
canoinoaaiiy iaoaiaoe/ieo iaoiaeia aeineiaeaeaiiy aeaoa?iiiiaaieo oa
noioanoe/ieo aaiethoeieieo nenoai» (iaeac N 25 aiae 20.01.97).

• IAe? «Eeaneoieaoeieii iaoiaee aeaaa?e, aiaeico oa aaiiao?i?» (aea?ae.
?a?no?. N 0197U019321), aeeiioaaii? ca eii?aeeiaoeieiith i?ia?aiith
Iiiinoa?noaa inaioe Oe?a?ie «Aaiiao?e/ii oa aiaeioe/ii iaoiaee a
iaoaiaoeoei oa ?? canoinoaaiiyo» (?ioaiiy iaoeiai-aenia?oii? ?aaee
Iiiinoa?noaa inaioe aiae 27.12.96, i?ioieie N 1).

Iniiaii iaoiaee aeineiaeaeaiiy — caaaeueiiaeaaa?a?/ii c aeei?enoaiiyi
iaoiaeia oai?i? iaiiaa?oi oa oai?i? iaeaeaeieaoeue.

Aaoi?ii cai?iiiiiaaii oaeiae iiai iaoiaee aea/aiiy no?oeoo?ieo
aeanoeainoae iaiiaa?oi aiaeiii?oiciia oa iaeaeaeieaoeue ia?aoai?aiue.
Iniiaith oeeo iaoiaeia ? aecia/aia aaoi?ii iaiiaa?oiiaa ocaaaeueiaiiy
iiiyooy a?oiiai? ia?e oa aaaaeaia a ?iaioi iiiyooy iaiia?ao?aeoei?
(eiai?, i?aai?, neiao?e/ii?) iiii?aeo. Ca aeiiiiiaith oeeo iiiyoue a
?iaioi iiaoaeiaaii iaiiaa?oiiai oa iaeaeaeieueoeaai eiino?oeoei?, a
oa?iiiao yeeo i noi?ioeueiaaii iniiaii ?acoeueoaoe.

Iniiaieie ?acoeueoaoaie ?iaioe ? oaa?aeaeaiiy oai?aoe/iiai ciinoo, yeee
i aecia/a? iaoeiao iiaecio ?iaioe.

Iniiaieie ?acoeueoaoaie ?iaioe ?:

1. iiaoaeiaaii eaoaai?ith iaiiaa?oiiaeo ia? oa eaoaai?ith cai?oie
iaiiaa?oiiai? ia?e, iienaii aieueii oa oiiaa?naeueii ia‘?eoe oeeo
eaoaai?ie;

eaoaai?iy iaiiaa?oiiaeo ia? ? ocaaaeueiaiiyi eaoaai?i? a?oiiaeo ia? A.
I.Ieioeiia; iienaiiy aieueieo ia‘?eoia oei?? eaoaai?i? ?
ocaaaeueithth/ith aiaeiiaiaeaeth ia ieoaiiy I.A.Eo?ioa («Oai?ey a?oii.
I.: Iaoea.- 1967», noi?.525) uiaei aoaeiae aieueieo ia‘?eoia eaoaai?i?
a?oiiaeo ia?; ocaaaeueiththoueny oaeiae oai?iy caaaeueieo aeiaooeia
Iiiia oa ?acoeueoaoe ?aaea? i Eiia i?i aoaeiao caaaeueiiai aeiaooeo
aeaio oeeeei/ieo a?oi;

2. iienaii no?oeoo?ii aeanoeainoi iaiiaa?oie aiaeiii?oiciia aieueii?
a?oie, ui aecia/athoue ?? c oi/iinoth aei iciii?oicio;

oea iienaiiy io?eiaii a oa?iiiao oai?i? uieueieo iaeaaeueieo ?icoe?aiue
i ? ocaaaeueiaiiyi ia iaeiiooaoeaiee aeiaaeie ?acoeueoaoia E.I.Aeoneiia
i?i iaiiaa?oie aiaeiii?oiciia eiiieieo i?inoi?ia oa iiaeoeia;

3. iiaoaeiaaii iiai iaiiaa?oiiai eiino?oeoei? (aiioeaaiai aieiii?oo,
aiaieiii?oo), a oa?iiiao yeeo iienaii aaeeoeaio (iao?e/io)
aeaeiiiiceoeith iaiiaa?oie aiaeiii?oiciia aieueii? a?oie;

eiino?oeoeiy aiioeaaiai aieiii?oo ocaaaeueith? eiino?oeoeith aiioeaaiai
aeiaooeo iaiiaa?oi;

4. iiaoaeiaaii eaoaai?ith NR -ni?yaeaiue a?oie oa iaeaeaeieueoey,
iienaii oiiaa?naeueii ia‘?eoe oei?? eaoaai?i?; a oa?iiiao aiaeiiaiaeii?
eiino?oeoei? NR -aeiaooeo a?oie oa iaeaeaeieueoey iienaii aoaeiao
neiao?e/iiai iaeaeaeieueoey ia aieueiie a?oii;

oeei iienaiiyi aeiiiaiththoueny oa iineeththoueny aiaeiii ?acoeueoaoe
i?i aoaeiao neiao?e/ieo iaeaeaeieaoeue, io?eiaii a ?iaioao Aa?iaia oa
Nieueaa?iaia, Iaeueae?oia oa ii., ?acoeueoaoe Oeiia?a i?i iaeaeaeieueoey
ia?aoai?aiue aieueieo a?oi;

5. ca aeiiiiiaith aiioeaaeo aieiii?oia iienaii aoaeiao iaiiaa?oi
aiaeiii?oiciia oeieeii 0-i?inoeo iaiiaa?oi;

oeei ?acoeueoaoii aeiiiaiththoueny ?acoeueoaoe E.I.Aeoneiia, I?anoiia,
Iaiia, Oaio?e i?i aiiiii?oicie oa eiia?oaioei? oeieeii 0-i?inoeo
iaiiaa?oi, ocaaaeueith?oueny oai?aia ?ina (Rees D. On semi-groups //
Proc. Cambridge Phil. Soc.- 1940.- 36.- p.387-400) i?i a?oio
aaoiii?oiciia oeieeii 0-i?inoi? iaiiaa?oie.

Oai?aoe/ia cia/aiiy ?acoeueoaoia ?iaioe aecia/a?oueny aianeii a
?ica‘ycaiiy i?iaeaie eeaneoieaoei? iaeaeaeieaoeue ia?aoai?aiue,
?icoe?aiiyi iaae canoinoaaiiy iaoiaea uieueieo ?icoe?aiue, iioe?aiiyi ia
oai?ith iaiiaa?oi oa oai?ith iaeaeaeieaoeue eiioeaioei? ni?yaeaiiy
aeaaa?a?/ieo nenoai, caeeaaeaii? oai?i?th A. I.Ieioeiia a?oiiaeo ia?, a
oaeiae oei, ui a ?iaioi io?eiaii ?ica‘ycie aiaeiieo caaea/ i?i aoaeiao
aeayeeo iaiiaa?oiiaeo oa a?oiiaeo eiino?oeoeie, ?icaeiaii oa
ocaaaeueiaii ?acoeueoaoe ?ina, E.I.Aeoneiia, ?aaea?, Eaai oa iioeo
aiaeiieo niaoeiaeinoia.

?acoeueoaoe ?iaioe ciaeaeooue canoinoaaiiy a aeineiaeaeaiiyo ui
caeieniththoueny iaoeiaeie eieaeoeaaie Ee?anueeiai, Eueaianueeiai,
Oa?eianueeiai, Aeiiaoeueeiai oiiaa?neoaoia, a iioeo inaioiio oa iaoeiaeo
caeeaaeao.

Ai?iaaoeith iniiaieo ?acoeueoaoia ?iaioe caeieniaii oeyoii ?o
iaaiai?aiiy ia

• IX Ananithciiio neiiicioii c oai?i? a?oi (Iineaa, aa?anaiue 1984),

• XIX Ananithciie aeaaa?a?/iie eiioa?aioei? (Eueaia, aa?anaiue 1987),

• Neai?nueeie oeiei c iiiaiaeaeia aeaaa?a?/ieo nenoai (Aa?iaoe, eeiaiue
1988),

• VII eiioa?aioei? «Algebra in logika» (NO?TH, Ia?iai?, /a?aaiue 1989),

• Iiaeia?iaeiie aeaaa?a?/iie eiioa?aioei?, i?enay/aiie iai‘yoi aeaae. A.
I.Iaeueoeaaa (Iiaineai?nuee, na?iaiue 1989),

• VI Neiiicioii c oai?i? eieaoeue, aeaaa? oa iiaeoeia (Eueaia, aa?anaiue
1990),

• Iiaeia?iaeiie aeaaa?a?/iie eiioa?aioei?, i?enay/aiie iai‘yoi i?io. A.
I.Oe?oiaa (Aa?iaoe, na?iaiue 1991),

• Iiaeia?iaeiie eiioa?aioei?, i?enay/aiie iai‘yoi aeaae. I.O.E?aa/oea
(Ee?a — Eooeuee, aa?anaiue 1992),

• IX Iiaeia?iaeiie eiioa?aioei? c oiiieiai? oa ?? canoinoaaiue (Ee?a,
aeiaoaiue 1992),

• III Iiaeia?iaeiie aeaaa?a?/iie eiioa?aioei?, i?enay/aiie iai‘yoi I.
I.Ea?aaiieiaa (E?aniiy?nuee, na?iaiue 1993),

• Iiaeia?iaeiie eiioa?aioei?, i?enay/aiie iai‘yoi I.A.*aaioa?ueiaa
(Eacaiue, /a?aaiue 1994),

• IV eiioa?aioei? «Groups and group rings» (Eueaia, aa?anaiue 1996),

• Iiaeia?iaeiie aeaaa?a?/iie eiioa?aioei?, i?enay/aiie iai‘yoi i?io.
E.I.Aeoneiia (Neia‘yinuee, na?iaiue 1997),

• Iiaeia?iaeiie aeaaa?a?/iie eiioa?aioei?, i?enay/aiie iai‘yoi i?io.
E.A.Eaeoaeiiia (Ee?a — Aiiieoey, o?aaaiue 1999),

• II Iiaeia?iaeiie eiioa?aioei? c oai?i? iaiiaa?oi (N.-Iaoa?ao?a,
eeiaiue 1999),

• aeaaa?a?/ieo naiiia?ao Ee?anueeiai, Eueaianueeiai, Oa?eianueeiai,
Aiiaeueneiai oiiaa?neoaoia 1985-1999 ??.

Ioaeieaoeith iniiaieo ?acoeueoaoia aeena?oaoei? caeieniaii a ?iaioao
aaoi?a [1-22], c yeeo 7 o niiaaaoi?noai.

?acoeueoaoe niiaaaoi?ia a aeena?oaoei? ia aeei?enoiaothoueny.

Ianya i no?oeoo?a ?iaioe. Ciino aeena?oaoeieii? ?iaioe aeeeaaeaii ia 291
noi?iioei iaoeiiieno i‘youeia ?icaeieaie, ui caaaeii iinoyoue 23
ia?aa?aoe. O anooii iaa?oioiaaii aeooaeueiinoue oaie aeena?oaoei?,
iaaaaeaii noeneee aeeeaae iniiaieo ?acoeueoaoia.

Ia?aeie eioa?aoo?e iinoeoue 175 iaeiaioaaiue.

2

CIINO ?IAIOE.

A ia?oiio ?icaeiei ?iaioe ?acii c iniiaieie iiiyooyie oai?i? iaiiaa?oi
oa iaeaeaeieaoeue iaaiaeyoueny ?acoeueoaoe aaoi?a i?i caaaeueii
aeanoeainoi iaiiaa?oi ia?aoai?aiue, iaiiaa?oi aiaeiii?oiciia aieueieo
a?oi, aeeno?eaooeaieo iaeaeaeieaoeue oa aeayeeo ?o ocaaaeueiaiue.
Iniiaieie ?acoeueoaoaie ?icaeieo ? aano?aeoia oa?aeoa?enoeea iaiiaa?oie
aiaeiii?oiciia aieueii? a?oie cei/aiiiai ?aiao oa iienaiiy
aeeno?eaooeaieo iieueiioaioieo iaeaeaeieaoeue c iaoaaaeueiaith
aaeeoeaiith a?oiith.

Aano?aeoio oa?aeoa?enoeeo iaiiaa?oie ?(X)=F(X) aiaeiii?oiciia aieueii?
a?oie F(X) a cei/aiiiio aeoaaioi X io?eiaii iaoiaeii E.I.Aeoneiia
iaeaaeueieo uieueieo ?icoe?aiue iaiiaa?oi. Aeey oeueiai aeei?enoaii
iiiaeeio ?0(X) onio oaeeo ???(X) , ia?ace yeeo ? oeeeei/ieie
iiaea?oiaie a?oie F(X) . Aeaiaioe iiiaeeie ?0(X) iacaaii a ?iaioi
aaoiiiaeaenoaaiiyie a?oie F(X) . Aeiaaaeaii, ui ?0(X) — uieueiee
iaeaae a ?(X) , a ?(X) — ?aeeia (c oi/iinoth aei iciii?oicio)
iaeneiaeueia uieueia ?icoe?aiiy iaeaaeo ?0(X) . Caiaene aeieeaa?
(oai?aia i.3.7, ?icae. I):

Iaiiaa?oia S oiaei e eeoa oiaei ? iciii?oiith iaiiaa?oii
aiaeiii?oiciia aeayei? aieueii? a?oie F(X) , eiee S ? iaeneiaeueiei
uieueiei ?icoe?aiiyi aeayeiai naiai iaeaaeo, iciii?oiiai iaiiaa?oii
?0(X) aaoiiiaeaenoaaiue a?oie F(X) .

Io?eiaii, e?ii oiai, iienaiiy iaeii?? c oaeoi?iaiiaa?oi iaiiaa?oie
?0(X) a oa?iiiao iaiiaa?oi ?ina iao?e/iiai oeio.

Iaeaeaeieueoea N c iaoaaaeueiaith aaeeoeaiith a?oiith iacaaii
iaeaeaeieueoeai Eaai, yeui aiii ? aeeno?eaooeaiei, aeiaooie xy
aoaeue-yeeo eiai aeaiaioia x,y?N iaeaaeeoue oeaio?o aaeeoeaii? a?oie i
xyz=0 aeey anio x,y,z?N (iieueiioaioiinoue noaiaiy 3). Aeia?a
aiaeiii, ui iaeaeaeieueoey Eaai aeieeathoue eiaeiiai ?aco, eiee ia
aeiaieueiie iaoaaaeueiaie a?oii (G,*) aecia/eoe ioeueoeieieaoeaio
iia?aoeith ca i?aaeeii xy=[x;y]=x*y*x* y . A aeiaieueiiio aeiaaeeo aeey
iienaiiy iaeaeaeieaoeue Eaai a ?iaioi aecia/a?oueny iiiyooy L -ia?e
iaoaaaeueiai? a?oie. L -ia?a (?,?) iaoaaaeueiai? a?oie G
neeaaea?oueny c aaaeueiai? a?oie ?G oa aiiiii?oicio ?: G?:x?x , ui
caaeiaieueiythoue aeayeei oiiaai, aeecueeei aei ? -oeaio?aeueiinoi a
?icoiiiii E.A.Eaeoaeiiia (Kaloujnine L. Ueber gewisse Beziehungen
zwischen einer Gruppen und ihren Automorphismen // Ber. Math.- Tagung
Berlin.- 1953.-164-172). Ia? iinoea oaa?aeaeaiiy (eaia i.4.13, oai?aia
i.4.14):

iaeaeaeieueoea N c iaoaaaeueiaith aaeeoeaiith a?oiith G oiaei e eeoa
oiaei ? iaeaeaeieueoeai Eaai, eiee inio? oaea L -ia?a (?,?) a?oie G
, ui xy=y?x*y aeey anio x,y?N .

Na?aae iioeo ?acoeueoaoia ia?oiai ?icaeieo aiaecia/eii iienaiiy
iaiiaa?oie oeueo?aaiaeiii?oiciia aeiaieueii? a?oie.

Ia?aoai?aiiy ? a?oie G iacaaii Un -aiaeiii?oiciii ( n2 ), yeui
(g1·…·gn)?=g1?·…·gn? aeey anio g1,…,gn?G . Aeiaaaeaii, ui
iiiaeeia UE(G) onio Un -aiaeiii?oiciia (aeey anio n?N , n2 ) a?oie
G ? iaiiaa?oiith aiaeiinii iia?aoei? eiiiiceoei?. Aeaiaioe iaiiaa?oie
UE(G) iacaaii a ?iaioi oeueo?aaiaeiii?oiciaie a?oie G . Eiee a?oia G
ia iinoeoue aeaiaioia neii/aiiiai ii?yaeeo ia?ii UE(G)=G . A
caaaeueiiio ae aeiaaeeo aeey iienaiiy aoaeiae iaiiaa?oi aiaeiii?oiciia
aecia/aii iiiyooy aoiiiiai iieiaiio, yea aeyaeeiny ei?eniei i aeey
iienaiiy iiaeiiii?aeia iaiiai?yiiai aeiaooeo iiii?aeia.

A §4 ia?oiai ?icaeieo ie?ii iaeaeaeieaoeue Eaai ?icaeyaeathoueny aeayei
caaaeueii aeanoeainoi iaeaeaeieaoeue. Iniiaiei i?e oeueiio aenooia?
iiiyooy ?aaeoeiaaiiai aiiiii?oicio a?oi.

Iaoae (G,*) , (H,*) — aeiaieueii a?oie, Q — iiaea?oia a?oie H ,
aeey yeeo aecia/aii aiaeia?aaeaiiy ?: GT(H):t?t, q:
GGQ:(x;y)(x;y)q. Aiaeia?aaeaiiy f: GH:ggf iacaaii ?aaeoeiaaiei
aiiiii?oiciii ic nenoaiith ?aaeoeoei? (?;q) (ei?iooa RQ}sigma
-aiiiii?oiciii), yeui (x*y)f=xf*yf?x*(x;y)q. O ?aci, eiee ?=idH ,
aiai?eoeiaii i?i RQ -aiiiii?oici. Ca iaaieo oiia RQ}sigma
-aiiiii?oicie iiaeooue aooe ioa?aeoa?eciaaieie ca aeiiiiiaith aaeeoeaii?
aeaeiiiiceoei? (oai?aia i.4.6, ?icae. I):

Iaoae U,H — a?oie, Q — iiaea?oia a?oie U , ?: H?U —
aioeaiiiii?oici. Aeaiaaeaioieie ? oaa?aeaeaiiy:

1. aiaeia?aaeaiiy f: H?U ? RQ}sigma -aiiiii?oiciii;

2. iniothoue a?oia ? , iiiiii?oici µ: U?? , RQµ -aiiiii?oici ?: H??
oa aioeaiiiii?oici ?: H?? oaei, ui f=(?)µ-1, ?h=µih?µ-1 i?e
aoaeue-yeiio h?H .

*a?ac ig ie iicia/a?ii aioo?ioiie aaoiii?oici a?oie, yeee
aecia/a?oueny ?? aeaiaioii g .

A ae?oaiio ?icaeiei ?iaioe iiaoaeiaaii iaiiaa?oiiaa ocaaaeueiaiiy
eaoaai?i? a?oiiaeo ia? A. I.Ieioeiia. Iiiyooy a?oiiai? ia?e aecia/aii A.
I.Ieioeiiei ye oi?iaeicaoeiy oeo /e iioeo aiaeiioaiue, ui aeieeathoue a
eeani ia‘?eoia eaoaai?i? a?oi. ?icaeoie oei?? iaea? i?e?iaeiuei aanoe a
aeaio iai?yieao — iiaoaeiaa caaaeueii? oai?i? ia? aeaaa?a?/ieo nenoai oa
niaoeiaeueii? oai?i? oaeeo ia?.

I?aaeiaoii caaaeueii? oai?i? ia? ? eaoaai?iy, ia‘?eoaie yei? ? ia?e
(A,B) , a ii?oiciaie ia‘?eoa (A1;B1) a ia‘?eo (A2;B2) — aiiiii?oicie
f:A1A2 , g:B1B2 , ui caaeiaieueiythoue oei /e iioei iaiaaeaiiyi, yei
aecia/athoueny ca‘yceaie iiae eiiiiiaioaie aiaeiiaiaeieo ia?. Oai?i?
iiaeeaiaeoaeueieo aeaaa?a?/ieo nenoai i?e oeueiio noathoue /anoeiith
caaaeueii? oai?i? aiaeiiaiaeieo ni?yaeaiue, eiee iaeio c eiiiiiaio ia?e
aaaaeaoe o?eaiaeueiith.

Aei i?aaeiaoii? iaeanoi niaoeiaeueii? oai?i? aiia?ieo ni?yaeaiue
aeaaa?a?/ieo nenoai neiae aiaeianoe ia?e (A;B) ?acii c aeayeeie
ii?oiciaie ?o eiiiiiaio a iioi aeaaa?a?/ii nenoaie. Iaei?inoioeie
ia‘?eoaie niaoeiaeueieo oai?ie ? cai?oee ia? aeaaa?a?/ieo nenoai —
aeiaa?aie aeaeyaeo

c ?icieie niiniaaie aecia/aiiy ?o ii?oiciia. *anoeiith niaoeiaeueii?
oai?i? ? eieaeueia oai?iy ia?e (A;B) , aea A , B — aeiaieueii (aea
oieniaaii) aeaaa?a?/ii nenoaie. Ii?oiciii aeuaiaaaaeaii? aeiaa?aie a
aeiaa?aio

i?e oeueiio aenooia? aiiiii?oici UU’ , ui caaeiaieueiy? i?e?iaeiii
aeiiaai uiaei eiiooaoeaiinoi eaaae?aoo, yeee i?e oeueiio aeieea?.

Ie?aio i?aaeiaoio iaeanoue (neacaoe a, iaeanoue eiino?oeoeaii? oai?i?
ia? aeaaa?a?/ieo nenoai) ooai?ththoue oiiaa?naeueii ia‘?eoe
niaoeiaeueieo eaoaai?ie oa eioiiaa?naeueii ia‘?eoe caaaeueieo eaoaai?ie
ia? c iaeyaeo ia ?o iiaeeeai canoinoaaiiy o aiaeiiaiaeieo no?oeoo?ieo
oai?iyo.

Ciino aaaaoueio aiaeiieo ia oeae /an ?acoeueoaoia aeicaiey? aiai?eoe i?i
oo /e iioo oi?io ?aaeicaoei? iioaioeiaeo caaaeueii? oa niaoeiaeueii?
oai?ie ia? aeaaa?a?/ieo nenoai, oi/a oi?ioaaiiy aiaeiiaiaeieo iai?yieia
a yaiiio aeaeyaei ua aeaeaei aiae caaa?oaiiy.

Iaeaieueo caaa?oaiiai aeaeyaeo iaaoea eeoa caaaeueia oai?iy a?oiiaeo
ia?, ii/aoie yei? aoei iieeaaeaii ?iaioaie A. I.Ieioeiia (Ieioeei A.E.
?aaeeeaeu a a?oiiiauo ia?ao // AeAI NNN?.-1961.-140.-n.1019-1022.) oa
E.A.Eaeoaeiiia (Kaloujnine L. Sur quelques proprietes des groupes
d’automorphismes d’un groupe abstrait // C. R.
Paris.-1950.-230.-2067-2069.). Iiaenoiie iaaiiai aoaio ?icaeoeo oai?i?
a?oiiaeo ia? aoei iiaeaaaeaii iiiia?aoi?th A. I.Ieioeiia (Ieioeei A.E.
A?oiiu aaoiii?oeciia aeaaa?ae/aneeo nenoai // I.:»Iaoea».-1966.-604n.),
a yeie iniiaio oaaao i?eaeieaii i?eeeaaeiiio aniaeoo oai?i? a?oiiaeo ia?
ye caniao aea/aiiy cia?aaeaiue a?oi a?oiaie aaoiii?oiciia aeaaa?a?/ieo
nenoai. ?acii c oei a oeie iiiia?aoi? aeeeaaeaii oni iaiaoiaeii aeaiaioe
caaaeueii? oai?i? a?oiiaeo ia? aae aei oai?i? ?aaeeeaeia a a?oiiaeo
ia?ao oa aiaeiiaiaeiinoae Aaeoa. Iiaeaeueoee ?icaeoie oai?i? a?oiiaeo
ia? iia‘ycaiee ia?o ca ana c ?iaioaie A. I.Ieioeiia (Ieioeei A.E.
A?oiiiaua iiiaiia?acey e iiiaiia?acey ia?, naycaiiuo n i?aaenoaaeaieyie
a?oii // Neae?neee iaoai.aeo?i.- 1972.-o. XIII, N 5.-n.1030-1053.), A.
I.Ieioeiia oa A.N.A?iiaa?aa (Ieioeei A.E., A?eiaa?a A.N. I iieoa?oiiao
iiiaiia?acee, naycaiiuo n i?aaenoaaeaieyie a?oii // Neae?neee
iaoai.aeo?i.-1972.-o. XIII, N 4.-n.841-858.), a yeeo a?oiiai ia?e
aea/athoueny ia ?iaii ?o iiiaiaeaeia. A. I.Ieioeiiei, cie?aia,
ocaaaeueiaii oai?aio Iaeiaiia-Oiaeueeiia i?i iaiiaa?oio iiiaiaeaeia
a?oi. *a?aiaee aoai ?icaeoeo oai?i? a?oiiaeo ia? aoei caaa?oaii
iiiia?aoi?th A. I.Ieioeiia oa N.I.Aiani (Ieioeei A.E., Aiane N.I.
Iiiaiia?acey i?aaenoaaeaiee a?oii. Iauay oai?ey, nayce e i?eeiaeaiey //
?eaa:Ceiaoia.-1983.).

Niaoeiaeueii oai?i? ia? aeaaa?a?/ieo nenoai iiee ui ia ooai?ththoue
naiinoieiiai iai?yieo, oi/a oaiiiaiieiai/ii oaei oai?i? aaea neeaeeny a
iaaeao aiiieiai/ii? aeaaa?e, oai?i? cia?aaeaiue oiui. Aeineoue
cacia/eoe, ui, iai?eeeaae, o?ae??iaa oai?iy a?oiiaeo ?icoe?aiue iiaea
aooe aeeeaaeaiith iiaith aiaeiiaiaeii? oai?i? o?ae??iaeo ia?, yei ia
aiaeiiio aiae a?oiiaeo ia?, ui aiaeiiaiaeathoue cia?aaeaiiyi,
aecia/athoueny ia?aiao?eciaaieie aiaeia?aaeaiiyie iaeii?? eiiiiiaioe a
a?oio aaoiii?oiciia ae?oai?.

Eaoaai?i? iiaeoeia iaae ?icieie eeanaie eieaoeue iinoyoue a niai
iaaecae/aeii oe?ieee niaeo? niaoeiaeueieo eaoaai?ie ia?.

Iaeoeiiaioei oaiiiaiii niaoeiaeueieo eaoaai?ie ? oaoiiea oiiaa?naeueieo
oa eioiiaa?naeueieo eaaae?aoia, canoinoaaiiy yei? iaiaeii?aciai
aeaiiino?oaaei ?? ieiaeiinoue (aeea., iai?eeeaae, Aaia?aeia A.E.
Ioiineoaeueiay aiiieiae/aneay aeaaa?a e ioiineoaeueiua a?oiiu
A?ioaiaeeea eieaoe // Aeenn. … aeieoi?a oec-iao iaoe.
Naieo-Iaoa?ao?a.-1991.-n.242.).

C?icoiiei, iaeiae, ui naiinoieiee ?icaeoie niaoeiaeueieo oai?ie ia?
aeaaa?a?/ieo nenoai ua /aea? iaiaoiaeieo noeioeia c aieo oai?ie
iiaeeaiaeoaeueieo aeaaa?a?/ieo nenoai, ui ia iiaeaeathoueny aea/aiith
iaoiaeaie eiiooaoeaii? aiiieiai/ii? aeaaa?e, oa iio?aao? ciinoiaii?
oaiiiaiieiai? ni?yaeaiue aeaaa?a?/ieo nenoai ?icieo eeania.
Iioaioeiaeueia iayaiinoue inoaiiuei? iaa?oioiao?oueny, cie?aia,
oai?aiith Ai?eaioa i?i ?aaeicaoeith a?oi a?oiaie aaoiii?oiciia
oiiaa?naeueieo aeaaa?, a oaeiae oai?aiith A?aioieooa-Iinoianueeiai i?i
?aaeicaoeith a?oi iiaea?oiaie a?oi aaoiii?oiciia iiaeaeae.

Aiaeiiaiaeii noeioee ca?iaeaeoaaeenue, cie?aia, a oai?i? iaiiaa?oi oa a
oai?i? iaeaeaeieaoeue.

Aeayei iiai aniaeoe niaoeiaeueii? oai?i? ia? aeieeathoue a ca‘yceo c
iiiyooyi aiiiii?oiciia cieaoaiiy (A?eai?/oe ?.E., Eo?/aiia I.O.
Iaeioi?ua aii?inu oai?ee a?oii, naycaiiua n aaiiao?eae // Eoiae iaoee e
oaoi. Nia?ai.i?iae.iaoai. Ooiaeai.iai?aaeaiey.-1990.- 58.- n.191-256.).

Ia ?aaeicaoeith iioaioeiaeo, ui noai?eany a oai?i? iaiiaa?oi oa a oai?i?
iaeaeaeieaoeue, a ae?oaiio ?icaeiei ?iaioe aecia/aii iiiyooy
iaiiaa?oiiai? ia?e oa niaoeiaeueii? eaoaai?i? cai?oie iaiiaa?oiiai?
ia?e. Iienaii oiiaa?naeueiee ia‘?eo oei?? eaoaai?i?. Aecia/aii oa
ioa?aeoa?eciaaii iiiyooy NR -ni?yaeaiiy a?oie oa iaeaeaeieueoey.

A iniiai iaoiaeia, ui canoiniaothoueny, eaaeeoue iiiyooy iaiia?ao?aeoei?
iiii?aeia, aaaaeaia oa aeaoaeueii aea/aia a ia?oiio ia?aa?aoi ?icaeieo.

Ia?aoai?aiiy ? iiii?aeo (M,*) iacaaii eiaith iaiia?ao?aeoei?th
oeueiai iiii?aeo, yeui (x*y)?=(x)? i?e aoaeue-yeeo x,y?M . A
aeai?noee niinia aecia/athoueny i?aai iaiia?ao?aeoei?. sseui
ia?aoai?aiiy ? iaeii/anii ? eiaith oa i?aaith iaiia?ao?aeoei?th, oi
aiai?eoeiaii i?i (neiao?e/io) iaiia?ao?aeoeith.

sseui ? — neiao?e/ia iaiia?ao?aeoeiy iiii?aeo (M,*) , oi ?? ia?ac ?
iiii?aeii aiaeiinii iia?aoei? a*}pi b= (a*b)? . Oeae iiii?ae
iicia/a?oueny /a?ac M}pi i iaceaa?oueny ? -iooaoei?th iiii?aeo M .
Eiaa (i?aaa) iaiia?ao?aeoeiy iiii?aeo M iaceaa?oueny ?aaoey?iith, yeui
?? ia?ac ? iiaeiiii?aeii a M . Io?eiaii e?eoa?i? ?aaoey?iinoi
iaeiiai/ieo iaiia?ao?aeoeie.

A §2 ae?oaiai ?icaeieo aecia/athoueny oa aea/athoueny iaiiaa?oiiai ia?e
A.Iaeiaia — ia‘?eoe caaaeueii? eaoaai?i? iaiiaa?oiiaeo ia?.

Ni?yaeaiiyi A.Iaeiaia iiii?aeia (M1,*) , (M2,*) iacaaii ia?o
aiaeia?aaeaiue ?: M1T(M2):t?t, ?: M2T(M1):t?t, ia?oa c yeeo ?
aiiiii?oiciii, ae?oaa — aioeaiiiii?oiciii ( T(X) — neiao?e/ia
iaiiaa?oia ia iiiaeeii X ), aeey yeeo aeeiiothoueny oiiae
(u*v)?x=u?x?x*v?x, x?M1,u,v?M2, (x*y)?u=x?u*y?u?x, x,y?M1,u?M2.
Noeoiiinoue (?: M1;M2:?) iaceaaoeiaii a oeueiio aeiaaeeo
iaiiaa?oiiaith ia?ith A.Iaeiaia. Cai?oeith ia?e A.Iaeiaia (?: M1;M2:?)
iaceaa?oueny iiii?aeia aeiaa?aia

ui caaeiaieueiy? oiiaai yr*xl=x?yl*y?xr, x?M1, y?M2. I?e?iaeiuei i?e
oeueiio aeieea? eaoaai?iy cai?oie ia?e A.Iaeiaia (?: M1;M2:?) .

Aoaeaii aiai?eoe, ui i?aaa iaiia?ao?aeoeiy ?1 oa eiaa iaiia?ao?aeoeiy
?2 iiii?aeo (M,*) ooai?ththoue aacen A.Iaeiaia [?1;?2]M oeueiai
iiii?aeo, yeui ?1*?2=?M, ?1?2=?2?1=?M, aea ?M — oioiaei?, a ?M —
ioeueiaa ia?aoai?aiiy. sseui [?1;?2]M — aacen A.Iaeiaia iiii?aeo M ,
oi iaiia?ao?aeoei? ?1 , ?2 ? ?aaoey?ieie.

Iiii?ae (M,*) iacaaii (M1;M2) -oaeoi?eciaiei ca A.Iaeiaiii, yeui
M1,M2 — iiaeiiii?aee iiii?aeo M oaei, ui M1?M2={?} i aoaeue-yeee
aeaiaio x?M iaeiicia/ii caieno?oueny o aeaeyaei x=x1*x2 , x1?M1 ,
x2?M2 .

Oeaio?aeueiei ?acoeueoaoii niaoeiaeueii? oai?i? iaiiaa?oiiaeo ia?
A.Iaeiaia ? oaa?aeaeaiiy (oai?aie ii.2.7, 2.8):

aeey iiii?aeia M,M1,M2 oa aiiiii?oiciia µ1: M1M , µ2: M2M
aeaiaaeaioieie ? oaa?aeaeaiiy:

1. aecia/aiith ? iaiiaa?oiiaa ia?a A.Iaeiaia (?: M1;M2:?) , aeey yei?
µ1,µ2 ? oiiaa?naeueiei ia‘?eoii ?? eaoaai?i? cai?oie;

2. inio? aacen A.Iaeiaia [?1;?2]M iiii?aeo M , aeey yeiai ?1µ1 ,
?2µ2 ;

3. iiii?ae M ? (M1µ1;M2µ2) -oaeoi?eciaiei ca A.Iaeiaiii.

Oiiaa?naeueiee ia‘?eo eaoaai?i? cai?oie iaiiaa?oiiai? ia?e A.Iaeiaia
(?: M1;M2:?) aecia/a?oueny iiii?aeii M=(?: M1M2:?) , ui aeieea? ia
iiiaeeii M1M2 , yeui iieeanoe (t;u)*(v;w)=(t*v?u;u?v*w),
t,v?M1,u,w?M2. Ii?oicie aiaeiiaiaeii? aeiaa?aie aecia/athoueny
aiaeia?aaeaiiyie µ1: M1M:ttµ1=(t;?), µ2: M2M:ttµ2=(?;t). Iiii?ae
M=(?: M1M2:?) iaceaa?oueny (?;?) -aeiaooeii A.Iaeiaia iiii?aeia M1
oa M2 . Eiee M1 , M2 — a?oie, io?eio?ii ciaiioith eiino?oeoeith
caaaeueiiai aeiaooeo oeeo a?oi. Niaoeiaeueia oai?iy iaiiaa?oiiaeo ia?
A.Iaeiaia ocaaaeueith? ?acoeueoaoe A.Iaeiaia, Oeaiia, Naia i?i aoaeiao
caaaeueieo aeiaooeia a?oi.

A §3 iiaoaeiaaii niaoeiaeueio oai?ith aecia/aieo ooo o?ae??iaeo
iaiiaa?oiiaeo ia?. ?acii c iiiyooyi o?ae??iai? iaiiaa?oiiai? ia?e
aeieea? eiino?oeoeiy o?ae??iaiai aeiaooeo (M1;M2)Sq?,? iiii?aeia M1 ,
M2 , aea ? , ? , q — nenoaia aiaeia?aaeaiue, ui ocaaaeueith?
iiiyooy nenoaie oaeoi?ia o?ae??iai? oai?i? a?oiiaeo ?icoe?aiue.

Iaiia?ao?aeoei?th ? iiii?aeo (M,·,e) iacaaii o?ae??iaith, yeui
M=e?-1·M? .

Ia? iinoea oai?aia (i.3.11):

aeey iiii?aeia M,M1,M2 aeaiaaeaioieie ? oaa?aeaeaiiy:

1. inio? o?ae??iaa ni?yaeaiiy [M1;M2]q?,? oaea, ui M(M1;M2)Sq?,? ;

2. inio? o?ae??iaa iaiia?ao?aeoeiy ? iiii?aeo M oaea, ui M1e?-1 ,
M2M}pi .

(Iaaaaea?ii, ui M}pi — ? -iooaoeiy iiii?aeo M — aeea. aeua).

Aoaeue-yea iaiia?ao?aeoeiy a?oie ? o?ae??iaith. Canoinoaaiiy
iaiia?ao?aeoeie aeicaiey? aeiaanoe, ui aoaeue-yeee aiaeiii?oici iooaoei?
aieueii? a?oie aoaeue-yeiai a?oiiaiai iiiaiaeaeo iiaeoeo?oueny
aiaeiii?oiciii oei?? a?oie.

A §4 ca aeiiiiiaith iaiia?ao?aeoeie oaoiieo iao?e/ii? aeaeiiiiceoei?
Ii?na ?iciianthaeaeaii ia iaiiaa?oie aiaeiii?oiciia aeiaooeia A.Iaeiaia.
Io?eiaii iaiiaa?oiiaee aiaeia oai?aie i?i iiey?iee ?iceeaae (oai?aia
i.4.14, ?icae. I)

A §5 ae?oaiai ?icaeieo iiaoaeiaaii niaoeiaeueio oai?ith NR -ni?yaeaiue
a?oie i iaeaeaeieueoey.

Iaoae N — iaeaeaeieueoea, H — a?oia [N*;H]q}sigma — o?ae??iaa
ni?yaeaiiy a?oi ( N* — aaeeoeaia a?oia iaeaeaeieueoey N ).
Aiaecia/eii, ui o?ae??iai ni?yaeaiiy a?oi aecia/athoueny o?ae??iaeie
nenoaiaie oaeoi?ia — /anoeiiei aeiaaeeii caaaeueiiai iiiyooy o?ae??iaiai
ni?yaeaiiy iiii?aeia. Aiaeia?aaeaiiy µ: HH?(NN; N):(x;y)µyx,
?:N?(HH;H):t?t iacaaii NR -ni?yaeaiiyi a?oie H oa iaeaeaeieueoey N
, yeui ((x1;y1)µy2x2;t1)µt2(x2;y2)?t1= (x1;(y1;t1)µt2y2)µ(y2;t2)?t1x2,
((x2;y2)?y1;t2)?t1=(x2;(y2;t2)?t1) ?(y1;t1)µt2y2,
(x1;y1*t1?y2*(y2;t2)q)µy2*t2x2=
(x1;y1)µy2x2*(x1;t1)µt2x2?(x2;y2)?y1* ((x2;y2)?y1;(x2;t2)?t1)q,
(x2;y2*t2)?y1*t1?y2*(y2;t2)q= (x2;y2)?y1*(x2;t2)?t1, i?e aoaeue-yeeo
x1,y1,t1?N , x2,y2,t2?H . sseui (µ;?) — aeayea NR -ni?yaeaiiy a?oie
H oa iaeaeaeieueoey N , oi a?oia (N*;H)Gq}sigma ia?aoai?th?oueny ia
iaeaeaeieueoea, yeui iieeanoe (x1;x2)(y1;y2)=((x1;y1)µy2x2;
(x2;y2)?y1).

Iaeaeaeieueoea, yea aecia/a?oueny ia (N*;H)Gq}sigma inoaiiueith
?iaiinoth iacaaii NR -aeiaooeii a?oie H oa iaeaeaeieueoey N , ui
aecia/a?oueny NR -ni?yaeaiiyi (µ;?) o?ae??iai? ia?e [N*;H]q}sigma
(aai, ei?iooa, NRµ,??,q -aeiaooeii a?oie H oa iaeaeaeieueoey N ).
NRµ,??,q -aeiaooie a?oie H oa iaeaeaeieueoey N iicia/aoeiaii /a?ac
(N;H)NRµ,??,q .

Ia?aoai?aiiy ? iaeaeaeieueoey N iacaaii ii?iaeueiith
iaiia?ao?aeoei?th oeueiai iaeaeaeieueoey, yeui ? ? iaiia?ao?aeoei?th
aaeeoeaii? a?oie iaeaeaeieueoey N i eiaith iaiia?ao?aeoei?th eiai
ioeueoeieieaoeaii? iaiiaa?oie.

NR -aeiaooee a?oi oa iaeaeaeieaoeue oa?aeoa?ecothoueny oaa?aeaeaiiyi
(i.5.7):

aeey iaeaeaeieaoeue N,N1 oa a?oie H aeaiaaeaioieie ? oaa?aeaeaiiy:

1. iniothoue o?ae??iaa ni?yaeaiiy [N1*;H]q}sigma oa NR -ni?yaeaiiy
(µ;?) , aeey yeeo N(N1;H) NRµ,??,q ;

2. inio? ii?iaeueia iaiia?ao?aeoeiy ? iaeaeaeieueoey N , aeey yei?
N1? , H(N*)}pi .

A o?aoueiio ?icaeiei niaoeiaeueia oai?iy iaiiaa?oiiaeo ia? A.Iaeiaia
aeiiiaith?oueny iniiaieie ?acoeueoaoaie caaaeueii? oai?i? i
canoiniao?oueny aei iiaoaeiae iiaeo iaiiaa?oiiaeo eiino?oeoeie, yei a
iiaeaeueoiio aeei?enoiaothoueny aeey iienaiiy aoaeiae aeayeeo iaiiaa?oi
aiaeiii?oiciia. Aeey a?oiiaeo ia? aaaaeaii iiiyooy aiaeiiaiaeiinoi a
a?oiiaie ia?i, ca aeiiiiiaith yeiai aaea?oueny iienoaaoe iiaea?oie
iaiiai?yieo oa i?yieo aeiaooeia a?oi.

Caaaeueio oai?ith iaiiaa?oiiaeo ia? A.Iaeiaia aeeeaaeaii a ia?oiio
ia?aa?aoi ?icaeieo. Ia‘?eoaie aiaeiiaiaeii? caaaeueii? eaoaai?i? ?
iaiiaa?oiiai ia?e A.Iaeiaia. Ii?oicie ia‘?eoa (?: M1,M2:?) a ia‘?eo
(: M1’, M2’:) aecia/a?oueny aiiiii?oiciaie f1: M1M1’:xxf1, f2:
M1M2’:xxf2, ui caaeiaieueiythoue oiiaai (t?v)f1=(tf1)vf2, t?M1,v?M2,
(v?t)f2=(vf2)tf1, t?M1,v?M2. Iniiaiei ?acoeueoaoii ? iienaiiy
aieueieo ia‘?eoia oei?? eaoaai?i?.

Aieueiee iiii?ae a aeoaaioi A iicia/aoeiaii /a?ac ?[A] . *a?ac l(w)
iicia/aoeiaii aeiaaeeio aeaiaioa w , a /a?ac w(k) — aeaiaio, ui a
caieno aeaiaioa w /a?ac oai?ii c A noi?oue ia k -io iinoei ( w(k)=1
, yeui k>l(w) ). Aeey anio i?N0 aecia/eii ia?aoai?aiiy si , fi
iiii?aeo ?[A] ca i?aaeeaie si(w)={

. Iaoae, aeaei, ?=?[X;Y] — aieueiee iiii?ae a aeoaaioi X?Y ( X?Y=
), ?X — i?aaee iaeaae aeaiaioia w?? , ui ii/eiathoueny c aeaiaioia
aeoaaioo X , ?Y — i?aaee iaeaae aeaiaioia w?? , ui ii/eiathoueny c
aeaiaioia aeoaaioo Y . *a?ac ?YX ( ?XY ) iicia/eii aieueiee iiii?ae a
aeoaaioi X(Y)={xv|x?X,v??Y} ( Y(X)={yt|y?Y,t??X} ). Aeey anio xv?X(Y)
, yt?Y(X) iieeaaeaii xv?yt=xy*v*t, yt?xv=yt*x*v. sseui w??YX ,
u??XY , oi, ca aecia/aiiyi ?w=?w(1)?w(2)…?w(l), l=l(w)
?u=?u(k)?u(k-1)…?u(1), k=l(u) w?a=sk-1(u)?a?u(k)* u(k)?a, a?X(Y),
w?b=w(1)?b*(w)fl-1 ?b?w(1), b?Y(X) Aeieea? i?e oeueiio iaiiaa?oiiaa
ia?a A.Iaeiaia (?:?YX,?XY:?) , yeo ie iaceaaoeiaii aieueiith
iaiiaa?oiiaith ia?ith A.Iaeiaia. Naia oaeeie ia?aie ae/a?io?oueny
aieueii ia‘?eoe eaoaai?i? iaiiaa?oiiaeo ia? A.Iaeiaia (oai?aia i.1.10,
?icae. III):

aieueii iaiiaa?oiiai ia?e i eeoa aiie ? aieueieie ia‘?eoaie eaoaai?i?
iaiiaa?oiiaeo ia? A.Iaeiaia.

Eino?oeoeiy aieueiiai ia‘?eoo eaoaai?i? iaiiaa?oiiaeo ia? ? aieueo
caaaeueiith iiae oi, yei canoiniaaii ?aaea? oa Eiiii aeey iiaoaeiae
caaaeueiiai aeiaooeo aeaio oeeeei/ieo a?oi.

A ae?oaiio ia?aa?aoi ?icaeieo aoaeothoueny eiino?oeoei? iao?e/iiai
niieo/aiiy i iiaeaieiiai iao?e/iiai niieo/aiiy iaiiaa?oiiaeo ia?. Aeey
oeueiai aeei?enoiaothoueny iaiiaa?oiiai ia?e (?: M1,M2:?) a eiaeiiio c
/anoeiieo aeiaaeeia, eiee iaeia c aiaeia?aaeaiue ?,? ? ioeueiaei.
Ia?oee oei ia? — (M1,M2:?) (ianeaii ceiaa), ae?oaee — (?: M1,M2)
(ianeaii ni?aaa).

Ia?e (?: T1,S) , (S,T2:?) , aea (S,*) , (T1,·) , (T2,·) —
iiii?aee, iacaaii ia?aie ic niieueiei iia?aiaeii S , yeui ?t?u=?u?t,
t?T1,u?T2. Ca oei?? oiiae iiiaeeia M=[T1 S;T2]={ (t;?s;u) t?T1,
s?S, u?T2}, aea ? — ciaiioiie ioeueiaee aeaiaio, ? iiii?aeii
aiaeiinii cae/aeii? iia?aoei? iao?e/iiai aeiaooeo, yeui iieeanoe
st=s?t, us=s?u, s?S,t?T1,u?T2. Iiii?ae M iaceaa?oueny iao?e/iei
niieo/aiiyi iaiiaa?oiiaeo ia? ic niieueiei iia?aiaeii. Io?eiaii
oiiaa?naeueio oa aioo?ioith oa?aeoa?enoeee iao?e/ieo niieo/aiue ia? ic
niieueiei iia?aiaeii. Io?eiaii oaeiae oa?aeoa?enoeeo a oa?iiiao
aecia/aieo a oeueiio ia?aa?aoi oeaio?aeueieo aacenia Ii?na.

Ia?e (?1: T,S1) , (?2: T,S2) ( S1,S2 — * -aaeeoeaii, T —
ioeueoeieieaoeaiee iiii?aee) iacaaii ia?aie ic niieueiei iia?aoi?iei
iiii?aeii. Aeey eiaeii? oaei? aeaieee ia? ei?aeoii aecia/aiei ?
iao?e/iee iiii?ae c aeaiaioaie (

), x?S1,t?T,y?S2 oa ic cae/aeiith iia?aoei?th iao?e/iiai aeiaooeo.
Oeae iiii?ae iacaaii iao?e/iei niieo/aiiyi ia? ic niieueiei iia?aoi?iei
iiii?aeii. Aeey oaeeo iao?e/ieo niieo/aiue oaeiae io?eiaii oiiaa?naeueii
oa aioo?ioii oa?aeoa?enoeee, a oaeiae oa?aeoa?enoeea a oa?iiiao
eioeaio?aeueieo aacenia Ii?na — iiiyooy, aeai?noiai aei iiiyooy
oeaio?aeueiiai aaceno.

Iaaaaeaii eiino?oeoei? iao?e/ieo niieo/aiue ii?aeiothoueny a
eiino?oeoei? iiaeaieiiai iao?e/iiai niieo/aiiy iaiiaa?oiiaeo ia?. Oey
eiino?oeoeiy aeieea?, yeui caaeaii ia?e (?: T,S1) , (S1,T1:?) ic
niieueiei iia?aiaeii S1 i ia?e (?: T,S2) , (S2,T2:?) ic niieueiei
iia?aiaeii S2 . Iiaeaieia iao?e/ia niieo/aiiy — iao?e/iee iiii?ae,
aeaiaioaie yeiai ? iao?eoei }sigma}psi}eta}varphi[

) x?T1,y?S1,t?T,u?S2,v?T2}. Io?eiaii oiiaa?naeueio, aioo?ioith
oa?aeoa?enoeee iaaaaeaieo eiino?oeoeie oa oa?aeoa?enoeeo a oa?iiiao oae
caaieo aacenia Ii?na, aecia/aieo a ?iaioi ca aeiiiiiaith nenoai
iaiia?ao?aeoeie, c iaaieie aeiaeaoeiaeie aeanoeainoyie.

A o?aoueiio ia?aa?aoi ?icaeieo iienaii iiaeiiii?aee oa eiia?oaioei?
eiino?oeoeie iao?e/iiai niieo/aiiy, yei iacaaii aoiiieie iiaeiiii?aeaie
oa aoiiieie eiia?oaioeiyie.

A §4 iiaoaeiaaii eieaeueio oai?ith a?oiiai? ia?e, aeoiaey/e c iiiyooy
aiaeiiaiaeiinoi ia oaeie ia?i oa cai?oee aiaeiiaiaeiinoi.

Iaoae (H1,H2:?) — aeayea a?oiiaa ia?a, aeey eiiiiiaio yei? aecia/aii
iiiiii?oicie µ1: U1H1:xxµ1, µ2: U2H2:xxµ2 i aiaeia?aaeaiiy f:
U2H1:xxf, yea caaeiaieueiy? oiiaai (x*y)f=(x;y)q*xf*(yf)?x, x,y?U2,
u?hihf?U1µ1, u?U1,h?H2, aea (x;y)q?U1µ1 (ca oa?iiiieiai?th §4
?icae. I aiaeia?aaeaiiy f ? ?aaeoeiaaiei aiiiii?oiciii a?oi). A
oeueiio aeiaaeeo aiai?eoeiaii, ui caaeaii aiaeiiaiaeiinoue [U1,U2:f] a
a?oiiaie ia?i (H1;H2:?) . Iiiaeeio U1

U2= {(u*tf;t)|u?U1,t?U2} iacaaii cai?oeith aiaeiiaiaeiinoi [U1,U2:f]
. Iniiaiith eaiith eieaeueii? oai?i? ? (eaie ii. 4.1, 4.2, ?icae. III):

Cai?oea aoaeue-yei? aiaeiiaiaeiinoi [U1,U2:f] a aoaeue-yeie a?oiiaie
ia?i (H1,H2:?) ? iiaea?oiith iaiiai?yiiai aeiaooeo H1}varphi
H2=(H1H2:?) . Aeey aoaeue-yei? iiaea?oie U a?oie G=H1}varphi H2
inio? aiaeiiaiaeiinoue [U1,U2:f] a a?oiiaie ia?i (H1,H2:?) oaea, ui
U=U1

Iiaeia?e a?oiiaeo ia? ? /anoeiiei aeiaaeeii aiaeiiaiaeiinoae a a?oiiaeo
ia?ao. ?acii c eiaeiith aiaeiiaiaeiinoth [U1,U2:f] a a?oiiaie ia?i
(H1,H2:?) aeieea? iiaeia?a (H1,U2:?) , i?e/iio [U1,U2:f]
caeeoa?oueny aiaeiiaiaeiinoth a oeie inoaiiie a?oiiaie ia?i.
Aiaeiiaiaeiinoue [U1,U2:f] a a?oiiaie ia?i (H1,H2:?) iacaaii
aieiaiith, yeui U2=H2 . Iaaeaei oni aiaeiiaiaeiinoi aaaaeaoeiooueny
aieiaieie.

Iiaea?oio U iaiiai?yiiai aeiaooeo G=H1}varphi H2 iacaaii oeieeii
?aaoey?iith, yeui U ? iiaeiaiiai?yiei aeiaooeii (a i?e?iaeiueiio
?icoiiiii) i U?H1 — ii?iaeueia iiaea?oia a?oie H1 . Aeey oeieeii
?aaoey?ieo iiaea?oi iaiiai?yieo aeiaooeia ia? iinoea ocaaaeueiaiiy
oai?aie Ooena i?i iiaei?yii aeiaooee aeaio a?oi (oai?aia i.4.5, ?icae.
III):

Iaoae G = H1 ? H2 . Aeey iiaeiaiiai?yiiai aeiaooeo U G a?oi H1 oa
H2 oaa?aeaeaiiy 1., 2., ui iaaiaeyoueny ieae/a, ? aeaiaaeaioieie:

1. U ? oeieeii ?aaoey?iith iiaea?oiith;

2. Iniothoue a?oia F , aioeaiiiii?oici ?: H2 ?F: x?x, aiiii?oici
?: H1 ?F: x x? oa nth?ue?eoeaiee no?auaiee aiiiii?oici ?: H2 ?F:
u u?, oaei, ui ?x?= ??x aeey anio x ?H2 i U = { (x; y) ?G x?=
y?}.

Canoinoaaiiy aiaeiiaiaeiinoae a a?oiiaeo ia?ao aeicaiey? io?eiaoe
iienaiiy iiaea?oi i?yiiai aeiaooeo aoaeue-yeiai neii/aiiiai /enea a?oi.

Iaoae G,? — a?oie, ? . R}triangle -aiiiii?oici (?aaeoeiaaiee
-aiiiii?oici) f: G? iacaaii eaacinth?ue?eoeaiei, yeui
?={x*gf|x?,g?G}.

Iaoae n2 — iaoo?aeueia /enei, In={1,2,?,n} , G={Gii?In} — niiaenoai
a?oi, N(G)={iGn-ii?In-1} . Ca iayaiinoth eaacinth?ue?eoeaieo Ri
-aiiiii?oiciia ?1: GnGn-1,?i: Gn-i+1 ?i-1i-1i-2Gn-i, 1 .
Aiaeiii?oicie Iaiaa?a a?oie G[x] aecia/athoueny noia?iiceoei?th neia
aiaeiiaiaeiiai aieueiiai aeiaooeo.

Aiaeiii?oici ? aieueiiai aeiaooeo aeaio a?oi iacaaii iaiiai?iaeoi?ii,
yeui ?1?=?1 , ?2??3=? , aea [?1;?2;?3] — aiaeiiaiaeiee aacen
o?ae??iaiai ?icuaieaiiy. Iiiaeeia onio iaiiai?iaeoi?ia ?
iiaeiaiiaa?oiith iaiiaa?oie aiaeiii?oiciia. Aeiaaaeaii, ui iao?e/ia
aeaeiiiiceoeiy iaiiaa?oie iaiiai?iaeoi?ia ? ieaeiueio?eeooiith. A?oia
G[x] aieiaei? aeayeei eaiiii/iei aacenii o?ae??iaiai ?icuaieaiiy.
Aeyaey?oueny, ui (oai?aia i.2.5, ?icae. IV):

iaiiaa?oia aiaeiii?oiciia Iaiaa?a a?oie G[x] niiaiaaea? c iaiiaa?oiith
iaiiai?iaeoi?ia a eaiiii/iiio aaceni o?ae??iaiai ?icuaieaiiy.

Canoinoaaiiy oaoiiee aiioeaaeo aiaieiii?oia aeicaiey? io?eiaoe ua iaeia
iienaiiy iaiiaa?oie aiaeiii?oiciia Iaiaa?a a?oie G[x] .

Iaoae ?x — iaeaiiioaioiee aiaeiii?oici a?oie G[x] ia , U=?x ,
? — aioeaiiiii?oici, ui noi?iaiaeaeo? aiaeiiaiaeiee iaiiai?yiee
?iceeaae, BH}sigma(U,) — eaaciaoiiiee iiaeiiii?ae aiaieiii?oo
BHWrU . *a?ac Mg(G[x]) iicia/eii iaiiaa?oio aiaeiii?oiciia Iaiaa?a
a?oie G[x] . Eaaciaoiiiee iiaeiiii?ae BH}sigma(U,) aecia/a?
iao?e/io aeaeiiiiceoeith iiii?aeo Mg(G[x]) (oai?aia i.2.6, ?icae. IV):

inio? iiiiii?oici Mg(G[x])BH}sigma(U,).

A o?aoueiio ia?aa?aoi /aoaa?oiai ?icaeieo iienaii aoaeiao iaiiaa?oie
aiaeiii?oiciia aieueii? a?oie neii/aiiai ?aiao.

Iaoae (q,?) — o?ae??iaa nenoaia oaeoi?ia, ui aecia/a? aieueio a?oio
Fn ?aiao n ye ?icoe?aiiy aieueii? a?oie F[Fn] ( [Fn] — eiiooaio
a?oie Fn ) ca aeiiiiiaith aieueii? aaaeueiai? a?oie Zn ?aiao n .
Iaiiaa?oia Fn io?eio? iienaiiy a oa?iiiao eaaciaoiiiiai iiaeiiii?aeo
aiaieiii?oo BHWrF(Zn) (oai?aia i.3.5, ?icae. IV):

ia? iinoea iciii?oici FnBHq}sigma(F,Zn)

A /aoaa?oiio ia?aa?aoi ?icaeieo iaaaaeaii iiaia iienaiiy
aaoiiiaeaenaoi?a aieueii? a?oie cei/aiiiai ?aiao.

Iicia/eii /a?ac M iiiaeeio onio NN -iao?eoeue µ=(µij), i,j?N , ui
caaeiaieueiythoue oiiaai:

1) µij?{-1;0;1}, i,j?N ;

2) aeey eiaeii? iao?eoei µ?M inio? L}mu?N oaea, ui eeoa L}mu
ia?oeo ?yaeeia iao?eoei µ ? iaioeueiaeie;

3) a eiaeiiio ?yaeeo aoaeue-yei? iao?eoei µ?M ia aieueoa iaeiiai
iaioeueiaiai aeaiaioo;

4) yeui µ=(µij?M , oi µijµi+1,j=-1 i?e aoaeue-yeeo i,j?N .

Iao?eoei, ui iaeaaeaoue iiiaeeii M iaceaaoeiaii M -iao?eoeyie. *enei
L}mu c oiiae 2) iaceaaoeiaii aenioith M -iao?eoei µ?M .

Aeey eiaeiiai iaoo?aeueiiai n /a?ac En* iicia/eii NN -iao?eoeth
(dij) oaeo, ui dij=-1 i?e i+j=n i dij=0 i?e i+j=n . sseui ??M,
L}mu=l , oi iieeaaeaii µ*=El*µ .

W -iao?eoeth µ=(µij) aenioith l=L}mu iaceaaoeiaii ia?iiaee/iith c
ia?aiao?ii ia?iiaee/iinoi P}mu=(p;q;r), p,q,r?N, r2 , yeui l=2(p-1)+qr
i aeeiiothoueny oiiae:

5) µij=µkj , k=p+sq+i-1 , 1sr-1 i?e pip+q-1 , j?N ;

6) yeui p>1 , oi µij=-µkj , k=p+qr-i+1 i?e 1ip-1 .

*a?ac W0 iicia/eii iiiaeeio onio iaia?iiaee/ieo W -iao?eoeue.

Iaoae, aeaei, V — iiiaeeia onio iaioeueiaeo oeiei/enaeueieo
iineiaeiaiinoae z=zii?N , ui iinoyoue i?eiaeiii iaeio neii/aiio
iineiaeiaiinoue /enae zi1,…, zik ( k?N ), yei ? aca?iiii?inoeie.
sseui z?V , µ=(µij)?W0 , oi /a?ac <µ;z> iicia/eii aaniethoio
aaee/eio /enea s=i(jµij)zi , iieeaaoe (µ;z)?={

. Iiiaeeia M={(z;d;µ)|z?V,d?N0, µ?W0} ? iaiiaa?oiith aiaeiinii
iia?aoei? (z(1);d1;µ(1))(z(2);d2;µ(2))=
(z(1);d1<µ(1);z(2)>d2;(El*)}xiµ(2)), (4.2) aea l=Lµ(2) ,
?=12(1+(µ(1);z(2))?) . Ooo /a?ac En*=(ij) iicia/aii oaeo NN
-iao?eoeth, ui ij=-1 , eiee i+j=n i ij=0 a i?ioeeaaeiiio aeiaaeeo.

Iiiaeeia M0={(z;0;µ)|z?V, µ?W0} ? iaeaaeii iaiiaa?oie M . *a?ac
M0(Z; N0;W0) iicia/eii oaeoi?iaiiaa?oio ?ina iaiiaa?oie M aiaeiinii
iaeaaeo M0 . Aeey aaoiiiaeaenaoi?a ?0(X) a?oie F(X) io?eio?ii oaea
iienaiiy (oai?aia i.4.4, ?icae. IV):

iaiiaa?oie ?0(X) oa M0(Z; N0;W0) ? iciii?oieie.

A §5 /aoaa?oiai ?icaeieo iienaii aoaeiao iaiiaa?oie aiaeiii?oiciia
aeiaieueii? oeieeii 0-i?inoi? iaiiaa?oie. Ooo aeyaeaii oiiaa?naeueiinoue
iaoiaeo iao?e/ii? aeaeiiiiceoei? ic canoinoaaiiyi aiioeaaeo aieiii?oia.

C eiaeiith ?aaoey?iith iaiiaa?oiith ?ina S=M0(I,G,J;P) i?e?iaeiuei
anioeith?oueny iiaeaieiee aiioeaaee aieiii?o WJI(G)=HWrJIG=[

], a yeiio naiaeai/ iao?eoey P aecia/a? aoiiiee iiaeiiii?ae
EMP}#(S) . Iaoae, aeaei, KJI(S) — aieiaia eiia?oaioeiy iiii?aeo
WJI(G)

Aoaeiaa iaiiaa?oie aiaeiii?oiciia S=M0(I,G,J: P) aecia/a?oueny
iciii?oiciii (oai?aia i.5.15): SEMP}#(S)/KJI(S).

A i‘yoiio ?icaeiei ?iaioe aea/athoueny iaeaeaeieueoey ia?aoai?aiue a?oi
oa aeayei iaeaeaeieueoeaai eiino?oeoei?. I?e oeueiio io?eiaii yeinii
iiaa ocaaaeueiaiiy aeeno?eaooeaieo iaeaeaeieueaoeue, ocaaaeueiaiiy
eiino?oeoei? i?yii? noie iaeaeaeieaoeue, aeiaaaeaii oai?aio aeai?noinoi
aeey iaeaeaeieaoeue O.Iaeiai aiaeiii?oiciia aieueieo a?oi oa oai?aio
aeaeiiiiceoei? neiao?e/ieo iaeaeaeieaoeue ia aieueieo a?oiao.

A §1 i‘yoiai ?icaeieo aecia/a?oueny oa aea/a?oueny iiiyooy ooieoeiiiaeo
ia a?oii.

Iaoae GX=(G;X) — a?oia onio aiaeia?aaeaiue iiiaeeie X a a?oio G .
X -ooieoeiiiaeii ia a?oii G (aai GX -ooieoeiiiaeii) iaceaaoeiaii
aiaeia?aaeaiiy f: GXG:??f. *a?ac (X;G)F iicia/eii a?oio onio GX
-ooieoeiiiaeia. sseui f?(X;G)F , oi eiai aaeeoeaiei aeaoaeoii iacaaii
aiaeia?aaeaiiy Af: GXGXG:(?1;?2)(?1;?2)Af, aea
(?1;?2)Af=(?1*?2)f*?2f* ?1f. Ia a?oiao GX i (X;G)F a i?e?iaeiie
niinia aecia/a?oueny no?oeoo?a NR -a?oi iaae neiao?e/iei
iaeaeaeieueoeai NT(G) (iaeaeaeieueoea ia?aoai?aiue a?oie G ). GX
-ooieoeiiiae f iacaaii E -iaeii?iaeiei, yeui f?=?f aeey anio ??G .
Iiiaeeia (X;G)F0 onio E -iaeii?iaeieo GX -ooieoeiiiaeia ?
iiaea?oiith a (X;G)F .

sseui f?(X;G)F0 , oi iiaea?oia Cf(G) a?oie G , ui ii?iaeaeo?oueny
iiiaeeiith Af ? ii?iaeueiith a G . Iiaea?oio Cf(G) iacaaii
aeaoaeoiith iiaea?oiith ooieoeiiiaeo f (aai f -aeaoaeoiith
iiaea?oiith), a iiaea?oio (G)f ui ii?iaeaeo?oueny iiiaeeiith F — f
-iiaea?oiith a?oie G . sseui iieeanoe G(X;f)={(u;?)|u?Cf(G),??GX},
DG(X;f)={(u;?)?G(X;f)|u=?f}, oi aiaeiinii iia?aoei?
(u1;?1)*(u2;?2)=(u1*u2i?1f* (?1;?2)Af;?1*?2) iiiaeeia G(X;f) ?
a?oiith c ii?iaeueiith iiaea?oiith DG(X;f) , yei caaeiaieueiythoue
oaa?aeaeaiith (i.1.5, ?icae. V):

(G)fG(X;f)/DG(X;f).

Iaoae F=F(X) — aieueia a?oia a aeoaaioi X , w=(x1,…,xn)w?F .
Aa?aaeueiei w -ooieoeiiiaeii (aai Vw -ooieoeiiiaeii) a?oie G iacaaii
GX -ooieoeiiiae µw , ui aecia/a?oueny oiiaith ?µw=(x1?1,…,xn?)w,
??GX. *a?ac (X;G)V iicia/eii iiiaeeio onio Vw -ooieoeiiiaeia, w?F
. (X;G)V — iiaea?oia a?oie (X;G)F0 . Aaeeoeaiee aeaoaeo Vw
-ooieoeiiiaeo µw iacaaii aa?aaeueiei w -eiiooaoi?ii (aai Vw
-eiiooaoi?ii) aeaiaioia ?1,?2?GX i iicia/eii /a?ac [?1;?2]w .
Aeaoaeoio iiaea?oio Vw -ooieoeiiiaeo µw?(X;G)V iacaaii aa?aaeueiei w
-eiiooaioii (aai Vw -eiiooaioii) i iicia/eii /a?ac [G]w . Vw
-eiiooaio a?oie caaaeaee iinoeoueny a ?? eiiooaioi. A?oio G iacaaii
Vw -aaaeueiaith, yeui [G]w={?} . Io?eio?ii ocaaaeueiaiiy iiiyooy n
-aaaeueiaeo a?oi (Aa?, E.A.Eaeoaeiii).

A?oio G iacaaii Z -i?yiei aeiaooeii nai?o ii?iaeueieo iiaea?oi
U1,…,Um , yeui G=U1*m , iiaea?oie U1,…,Um ? iiia?ii iiaeaiaioii
ia?anoaaieie i i?e aoaeue-yeiio 1im ia?aoei Ui?(U1*i-1*Ui+1*m)
iinoeoueny a oeaio?i a?oie G .

Iaoae G — Vw -aaaeueiaa a?oia, {xij}j=1k — aieueii oai?ii, ui
aoiaeyoue aei caieno aeaiaioa w?F c iaioeueiaeie noiaie iieacieeia.
*a?ac ?i iicia/eii iaeaiiioaioii aiaeiii?oicie a?oie F c ?i= .
Iiaea?oie µw?ij , 1jk iacaaii aieiaieie w -eiiiiiaioaie a?oie G .
Iineeaiiyi oai?aie i?i ?icoaooaaiiy aa?aaeueii? iiaea?oie, ui
ii?iaeaeo?oueny iaeiei neiaii (A?thi), oa aeiiiaiaiiyi aei oai?aie
E.A.Eaeoaeiiia i?i aoaeiao n -aaaeueiaeo a?oi ? oaa?aeaeaiiy (i.1.11,
?icae. V):

aa?aaeueia w -iiaea?oia aoaeue-yei? Vw -aaaeueiai? a?oie ? Z -i?yiei
aeiaooeii ?? aieiaieo w -eiiiiiaio.

A §2 ?icaeieo aea/athoueny iaeaeaeieueoey, ui aeieeathoue ?acii c
iiiyooyi aa?aaeueiiai aiaeiii?oicio.

Ia?aoai?aiiy ??NT(G) iacaaii aa?aaeueiei w -aiaeiii?oiciii (aai Vw
-aiaeiii?oiciii) a?oie G , yeui µw?=?µw.

Iiiaeeio onio Vw -aiaeiii?oiciia a?oie G iicia/eii /a?ac Ew(G) . O
aeiaaeeo, eiee G ? w -aaaeueiaith iiiaeeia Ew(G) ?
iiaeiaeaeaeieueoeai neiao?e/iiai iaeaeaeieueoey NT(G) — neiao?e/iei
iaeaeaeieueoeai NEw(G) Vw -aiaeiii?oiciia w -aaaeueiai? a?oie G .
Iienaii ca‘ycee, yei iniothoue iiae w -aeeno?eaooeaieie
iaeaeaeieueoeyie oa iaeaeaeieueoeyie NEw(G) .

A §3 i‘yoiai ?icaeieo iienaii eiino?oeoeith aecia/aii? ooo
iaiiaaeeno?eaooeaii? noie iaeaeaeieaoeue.

Iaoae N1 , N2 — iaeaeaeieueoey c iaeeieoeyie e1 i, aiaeiiaiaeii,
e2 . sseui aecia/aii aioeaiiiii?oici ?: M(N2)N1:t?t
ioeueoeieieaoeaii? iaiiaa?oie M(N1) iaeaeaeieueoey N2 , oi iia?aoei?
(t1;u1)*(t2;u)=(t1*t2;u1*u2), t1,t2?N1,u1,u2?N2,
(t1;u1)(t2;u2)=(t1·t2?u1;u1u2), t1,t2?N1,u1,u2?N2 ia?aoai?ththoue
iiiaeeio N1N2 ia iaeaeaeieueoea, ui iacaaia a ?iaioi
iaiiaaeeno?eaooeaiith noiith iaeaeaeieaoeue N1 oa N2 i iicia/aii
/a?ac (N1;N2)?}sigma .

?ao?aeoi?ii iaeaeaeieueoey N c iaeeieoeath e iacaaii ia?o
i?oiaiiaeueieo iaeaiiioaioia ’,?N , ui caaeiaieueiythoue oiiaai
’*=’=e, x=(x’*)(’*x), x?N, — aeeno?eaooeaiee.
Iaiiaaeeno?eaooeaii noie ciaoiaeyoue naith oa?aeoa?ecaoeith a oa?iiiao
?ao?aeoi?ia (oai?aia i.3.8, ?icae. V):

aeey iaeaeaeieaoeue N,N1,N2 aeaiaaeaioieie ? oaa?aeaeaiiy:

1. N=(N1;N2)?}sigma ;

2. inio? ?ao?aeoi? iaeaeaeieueoey N , yaea?ia iaeaeaeieueoea yeiai ?
iciii?oiei iaeaeaeieueoeth N1 , a aeoeaiee ia?ac — iaeaeaeieueoeth N2
.

ssaea?iei iaeaeaeieueoeai ?ao?aeoi?a (’;) i?e oeueiio iaceaa?ii
iaeaeaeieueoea, ui aecia/a?oueny ia N’ ioeueoeieieaoeaiith iia?aoei?th
x’?x’= (x’*)x2’ , x1,x2?N , a aeoeaiei ia?acii — iaeaeaeieueoea N .

Iaiiaaeeno?eaooeaii noie iienothoueny oaeiae a oa?iiiao neiao?e/ieo
iaeaeaeieaoeue ia i?yieo aeiaooeao a?oi.

A §4 i‘yoiai ?icaeieo aea/athoueny oae caaii iaeaeaeieueoey O.Iaeiai —
iaeaeaeieueoey, ui aecia/athoueny ia iiiaeeii onio aiaeiii?oiciia
aieueii? a?oie.

Iaoae E=F(X) — iaiiaa?oia aiaeiii?oiciia aieueii? a?oie F=F(X) a ia
aieueo iiae cei/aiiiio aeoaaioi X . sseui aeey ?1,?2?E /a?ac ?1*?2
iicia/eoe aiaeiii?oici ? oaeee, ui x?=x?1*x?2 aeey anio x?X , oi
iaiiaa?oia E ia?aoai?th?oueny ia iaeaeaeieueoea NE(F) , yea ie
iaceaa?ii iaeaeaeieueoeai O.Iaeiai aieueii? a?oie F(X) .

Iaoae FX — a?oia onio aiaeia?aaeaiue iiiaeeie X a a?oio F=F(X) .
Aecia/eaoe aca?iii iaa?iaii ai?eoei? d:FFX:?d}varphi=?|X, E:
FXF:?E}sigma io?eio?ii iiaeeeainoue ia?aoai?eoe a?oio FX ia
iaeaeaeieueoea, iieeaaoe ?1?2=dE?1E?2 , ?1,?2?FX . Iacaaii oea
iaeaeaeieueoea iaeaeaeieueoeai X -iineiaeiaiinoae a?oie F i iicia/eii
/a?ac NS(X;F) . Ia? iinoea oai?aia aeai?noinoi (i.4.7, ?icae. V):

iaeaeaeieueoea O.Iaeiai ? iciii?oiei iaeaeaeieueoeth X -iineiaeiaiinoae
ia a?oii F : NE(F(X))NS(X;F).

sseui aeey anio ??NT(F) /a?ac ?’ iicia/eoe aiaeiii?oici a?oie F
oaeee, ui x?=x?’ , x?X , oi ia?aoai?aiiy ?:??}tau=?’ aeyaey?oueny
ii?iaeueiith iaiia?ao?aeoei?th neiao?e/iiai iaeaeaeieueoey NT(F) . Oea
aeicaiey? io?eiaoe oai?aio aeaeiiiiceoei? neiao?e/iiai iaeaeaeieueoey
NT(F) (i.4.2, ?icae. V):

neiao?e/ia iaeaeaeieueoea NT(F) ia aieueiie a?oii F ? NR -aeiaooeii
iaeaeaeieueoey O.Iaeiai NE(F) oa a?oie X -aioeyoi?ia
U(X)={??NT(G)|x?=?,x?X}.

Eiino?oeoeiy NR -aeiaooeo, oaeei /eiii, aeicaiey? iiaeieaoe o?oaeiiui,
coiiaeaii i?inoioith neiao?e/ieo iaeaeaeieaoeue oa iaiaaeaiinoth a
ca‘yceo c oeei caaaeueiiaeaaa?a?/ieo oaeoi?ecaoeieieo iaoiaeia.

Aeniiaee. Iniiaii ?acoeueoaoe ?iaioe noinothoueny i?iaeaie eeaneoieaoei?
iaeaeaeieaoeue ia?aoai?aiue ca aeanoeainoyie ?o ioeueoeieieaoeaieo
iaiiaa?oi. A ?iaioi ?icaeiooi iiai iaoiaee, ui ocaaaeueiththoue aiaeiii
iaoiaee oai?i? iaiiaa?oi aiaeiii?oiciia iaoaiaoe/ieo no?oeoo? (a
?icoiiiii Ao?aaei) oa oai?i? a?oiiaeo ia?, iiaoaeiaaii iiai aeaaa?a?/ii
eiino?oeoei?. Cai?iiiiiaaii iaoiaee oa eiino?oeoei? canoiniaaii aei
iienaiiy no?oeoo?ieo aeanoeainoae iaeaeaeieaoeue ia?aoai?aiue oa ?o
ioeueoeieieaoeaieo iaiiaa?oi.

A oeieiio ?iaioith aecia/aii iiaee iai?yiie — ioeueoeieieaoeaia oai?iy
iaeaeaeieaoeue ia?aoai?aiue. ?? ?acoeueoaoe ? aianeii a ?ica‘ycaiiy
aeooaeueii? i?iaeaie no?oeoo?ii? eeaneoieaoei? iaeaeaeieaoeue
ia?aoai?aiue oa ?o ioeueoeieieaoeaieo iaiiaa?oi.

?iaioe aaoi?a ca oaiith aeena?oaoe??.

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Oace aeiiia?aeae.

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26. Usenko V.M. Semidirect products and monoids of semilinear
transformations // 7 konferenca Algebra in logika (Maribor, junij
1989).- Maribor.- 1989.- p.31-33.

27. Onaiei A.I. Ne?auaiiua aiiiii?oeciu e iieoi?yiua i?iecaaaeaiey //
Iaaeaeoia?iaei. eiio. ii aeaaa?a (Iiaineae?ne, aaaono 1989).
Oac.aeiee.- Iiaineae?ne: Ei-o iaoai. NI AI NNN?.- 1989.- n.126.

28. Ee?e/aiei A.A., Onaiei A.I. I iieoi?yiuo i?iecaaaeaieyo
ii/oeeieaoe // VI neii. ii oai?ee eieaoe, aeaaa? e iiaeoeae (Eueaia,
naioya?ue 1990). Oac.niiau.- Eueaia: Eueaia.ain.oi-o.- 1990.- n.67.

29. Ee?e/aiei A.A., Onaiei A.I. Noaaaeaeeoeaiua eaoaai?ee e
ii/oeeieueoea // Iaaeaeoia?iaei. eiio. ii aeaaa?a (Aa?iaoe, aaaono
1991). Oac. aeiee. ii oai?ee eieaoe, aeaaa? e iiaeoeae.- Iiaineae?ne:
Ei-o iaoai. NI AI NNN?.- 1991.- n.121.

30. Onaiei A.I. Ia aaaeaauo ii/oeeieueoeao // Oace i?aeia?iaei.
eiio., i?enay/aii? iai`yo? aeaae. I.I.E?aa/oea (Ee?a-Eooeuee, aa?anaiue
1992).- Ee?a:?i-o iaoai. AI Oe?a?ie.- 1992.- n.221.

31. Onaiei A.I. Eiaa?eaioiua naoe a aeeioaioiaie aaiiao?ee e
ii/oeoeeueo?u // Oacenu IX iaaeaeoia?iaei. eiio. ii oiiieiaee e aa
i?eeiaeaieyi (Eeaa, naioya?ue 1992).- Eeaa: Ei-o iaoai. AI Oe?aeiu.-
1992.- n.44.

32. Onaiei A.I. Ii/oeeieueoea aa?aaeueiuo yiaeiii?oeciia // III
iaaeaeoia?iaei. eiio. ii aeaaa?a iaiyoe I.E.Ea?aaiieiaa. (E?aniiy?ne,
aaaono 1993). Oac.aeiee.- E?aniiy?ne: NI ?IAI.- 1993.- n.341.

33. Onaiei A.I. O-eiiiooaiou a?oii e aeeno?eaooeaiua ii/oeeieueoea //
Aeaaa?a e aiaeec. Oac. aeiee. iaaeaeoia?iaei. iao/iie eiio.,
iinayuaiiie 100-eaoeth ni aeiy ?iaeae. I.A.*aaioa?aaa (Eacaiue, ethiue
1994). *.1.- Eacaiue: Eac.Oi-o.- 1994.- n.95.

34. Onaiei A.I. Yiaeiii?oeciu naiaiaeiuo a?oii // I?aeia?iaeia
aeaaa?. eiio. i?enay/. iai`yo? i?io. E.I.Aeone?ia. (Neia`yinuee,
na?iaiue 1997).- Ee?a: ?i-o iaoai. IAI Oe?a?ie.- 1997.- n.22.

35. Onaiei A.I. I eaoaai?ee iieoa?oiiiauo ia? A.Iaeiaia // Ae?oaa
i?aeia?iaeia aeaaa?. eiio. a Oe?a?i?, i?enay/aia iai`yo? i?io.
E.A.Eaeoaei?ia. (Ee?a-A?iieoey, o?aaaiue 1999).- A?iieoey:AAeIO.- 1999.-
n.118-120.

36. Onaiei A.I. Iieo?ao?aeoeee iiiieaeia // Ibid.- c.120-121.

37. Usenko V.M. On endomorphisms of free groups // II
Iaaeaeoia?iaei. eiio. ii oai?ee iieoa?oii (N.-Iaoa?ao?a, etheue 1999).-
N.Iaoa?ao?a: «Naaa?iue i/aa».- 1999.- n.81.

Onaiei A.I. Iai?aa?oie oa iaeaeae?eueoey ia?aoai?aiue.-?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaai nooiaiy aeieoi?a o?ceei-iaoaiaoe/ieo
iaoe ca niaoe?aeuei?noth 01.01.96 – aeaaa?a oa oai??y /enea. –
Ee?anueeee iaoe?iiaeueiee oi?aa?neoao ?iai? Oa?ana Oaa/aiea, Ee?a.

Caoeua?oueny 22 iaoeiaeo ?iaioe, ui i?noyoue ?acoeueoaoe c oai???
iai?aa?oi oa iaeaeae?eaoeue ia?aoai?aiue.

A ?iaio? ae?oo?thoueny ieoaiiy eeaneo?caoe?? iaeaeae?eaoeue ia?aoai?aiue
ca aeanoeainoyie ?o ioeueoeie?eaoeaieo iai?aa?oi.

Aecia/aii iai?aa?oiia? oa iaeaeae?eueoeaa? eiino?oeaoe??, a oa?i?iao
yeec iienaii on?ooe?oi? aeanoeaino? aeayeeo eean?a iai?aa?oi
aiaeiii?c?ci?a oa iaeaeae?eaoeue ia?aoai?aiue.

Eeth/ia? neiaa: iai?aa?oia, iaeaeae?eueoea, iai?aa?oia aiaeiii?o?ci?a,
iaeaeae?eueoea ia?aoai?aiue, iai?a?aoiiaa ia?a, iai?aa?oiiaa
eiino?oeoe?y, no?ooeo?i? aeannoeaino? iai?aa?oi oa ieaeaeae?eaoeue
ia?aoai?aiue.

Onaiei A.I. Iieoa?oiiu e ii/oeeieueoeu i?aia?aciaaiee.-?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
iaoeeiiaeueiue oieaa?neoao eiaie Oa?ana Oaa/aiei, Eeaa.

Caueuaaony 22 iao/iua ?aaiou, niaea?aeauea ?acoeueoaou ii oai?ee
iieoa?oii e ii/oeeieaoe i?aia?aciaaiee.

A ?aaioa ?aoathony caaea/e eeanneoeeaoeee ii/oeeieaoe i?aia?aciaaiee n
iiiiuueth naienoa eo ioeueoeieeeaoeaiuo iieoa?oii.

Ii?aaeaeaiu iieoa?oiiiaua e ii/oeeieueoeaaua eiino?oeoeee, a oa?ieiao
eioi?uo iienaiu no?ooeo?iua naienoaa iaeioi?uo eeannia iieoa?oii
yiaeiii?oeciia e ii/oeeieaoe i?aia?aciaaiee.

Eeth/aaua neiaa: iieoa?oiiu, ii/oeeieueoea, iieoa?oiiu yiaeiii?oeciia,
iio/eeieueoea i?aia?aciaiee, iieoa?oiiiaua ia?u, iieoa?oiiiaua
eiino?oeoeee, no?oeoo?iua naienoaa iieoa?oii e ii/oeeieaoe
i?aia?aciaaiee.

Usenko V.M. Semigroupsand near-rings of transformations. – Manuscropt.

Thesis for a doctor’s degree by speciality 01.01.06 – Algebra and theory
of numbers.-Kyiv Taras Shebchenko University, Kyiv.

22 scientific papers, that contrained the results in the thory of
transformation semigroups and in the theory near-rings of
transformations are defended.

The classification problems of near-rings of transformations by the
properties of its multiplicative semigroups are solved.

The semigroups and the near-ring constructions are defined for
describing of structure properties of some classes encomorphisms
semigpoups and transformations near-rings.

Key words: semigroups, near-ring. Endomorphisms semogroups,
transformations near-rings, semigroups pair, semigroups construction,
structures properties of semigroups and near-rings of transformations.

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