Вільні групи та напівгрупи автоматних перетворень: Автореф. дис.. канд. фіз-мат. наук / А.С. Олійник, Київ. ун-т ім. Т.Шевченка. — К., 1999. — 16 с. —

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

Ie?eiee Aiae??e Noaiaiiae/

OAeE 512.53/54

A?EUeI? A?OIE OA IAI?AA?OIE

AAOIIAOIEO IA?AOAI?AIUe

(01.01.06 — aeaaa?a oa oai??y /enae)

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Ee?a 1999

Aeena?oaoe??th ? ?oeiien.

?iaioo aeeiiaii a Ee?anueeiio oi?aa?neoao? ?iai? Oa?ana Oaa/aiea.

Iaoeiaee ea??aiee:

NOUAINUeEEE A?oae?e ?aaiiae/, aeieoi? o?ceei-iaoaiaoe/ieo iaoe,
i?ioani?, caa?aeoaa/ eaoaaepith aeaaape ? iaoaiaoe/ii? eia?ee
Ee?anueeiai oi?aapneoaoo ?iai? Oapana Oaa/aiea, i.Ee?a

Io?oe?ei? iiiiaioe:

NENAE ss?ineaa I?ieiiiae/, aeieoi? o?ceei-iaoaiaoe/ieo iaoe, ipia?aeiee
iaoeiaee ni?apia?oiee ?inoeoooo iaoaiaoeee HAH Oepa?ie, i. Ee?a

E?IOEI Iaoae?y A?oae??aia, eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe,
aeeeaaea/ Oe?a?inueeiai aea?aeaaiiai oi?aa?neoaoo oa?/iaeo oaoiieia?e,
i.Ee?a

I?ia?aeia onoaiiaa:

Euea?anueeee aea?aeaaiee oi?aa?neoao ?iai? ?aaia O?aiea, i.Euea?a

Caoeno a?aeaoaeaoueny « 30 » na?iiy 1999 ?ieo i 14 aiae.
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?. aeaae. Aeooeiaa, 6, 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 «22» /a?aiy 1999 ?ieo.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee
EE?E*AIEI A.A.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Na?aae on?o oei?a caaea/ i?i a?euei? a?oie iniaeeaa
i?noea iin?aeathoue caaea/? iiaoaeiae eiie?aoieo cia?aaeaiue oaeeo a?oi.
Ine?eueee a?eueia a?oia ?aiao aeaa i?noeoue a?eueio i?aea?oio ce?/aiiiai
?aiao, a oiio ? a?euei? i?aea?oie aeia?eueieo ne?i/aiieo ?aia?a, oi, ye
i?aaeei, ?icaeyaeathoue cia?aaeaiiy oeie /e ?ioeie ia’?eoaie naia
a?eueii? a?oie ?aiao aeaa. Aeia?a a?aeii? cia?aaeaiiy Iaaiona oe???
a?oie oi?iaeueieie noaiaiaaeie ?yaeaie iaae e?eueoeai oe?eeo /enae a?ae
ci?iieo, ye? ia eiiooothoue, ca aeiiiiiaith yeiai aoei ioa?aeoa?eciaaii
?? ieaei?e oeaio?aeueiee ?yae, aai cia?aaeaiiy Naiiaa iaae?iaeaeaieie
oe?ei/enaeueieie iao?eoeyie ae?oaiai ii?yaeeo, yea aea? ciiao
anoaiiaeoe, ui a?euei? a?oie ai?ieneiothoueny ne?i/aiieie p-a?oiaie
aeey aeia?eueiiai i?inoiai p.

?nioaaiiy cia?aaeaiiy a?eueii? a?oie ?aiao 2 i?aenoaiiaeaie
iane?i/aiiiai noaiaiy aa?aioo?oueny oai?aiith Eae? i?i cia?aaeaiiy a?oi
i?aenoaiiaeaie. ?acii c oei, a e?eueeio ?iaioao inoaii?o ?ie?a
aoaeothoueny ?? aeniei o?aiceoeai? i?aenoaiiaeia? cia?aaeaiiy (a?oia
i?aenoaiiaie iane?i/aiiiai noaiaiy iaceaa?oueny aeniei o?aiceoeaiith,
yeui aiia ? k-o?aiceoeaiith aeey eiaeiiai iaoo?aeueiiai k).

, ii?iaeaeothoue a?eueio a?oio.

Iaeei c ooiaeaiaioaeueieo ?acoeueoao?a i?i a?euei? a?oie, a?aeiiee i?ae
iacaith aeueoa?iaoeaa O?ona aia?oa aeiaiaeeoueny a i?ea?iaeuei?e
?iaio? o oaeiio oi?ioethaaii?:

Iaae iieai oa?aeoa?enoeee 0 aeia?eueia e?i?eia a?oia aai i?noeoue
iaaaaeaao a?eueio i?aea?oio, aai a ia? ? ?ica’ycia i?aea?oia ne?i/aiiiai
?iaeaena.

Oeae ?acoeueoao aoa iaaiei iiooiaoii o iai?yieo aeine?aeaeaiiy ia i?inoi
ie?aieo cia?aaeaiue a?eueii? a?oie, a iiaaae?iee iiiaeeie a?eueieo
i?aea?oi na?aae on?o i?aea?oi o??? /e ?ioi? a?oie. C’yaeeany na??y
?acoeueoao?a, o yeeo noaa?aeaeo?oueny, ui a oiio /e ?ioiio ?icoi?ii?,
«a?eueo?noue» i?aea?oi aeaii? a?oie ? a?eueieie. A yeiio nain?
aaeeaa?oueny ooo neiai «a?eueo?noue»?

) aeaea?o?a noai?iue oe??? a?oie ? aeiaiaeeoueny, ui iiiaeeia oeo
aeaiaio?a oeueiai noaiaiy, eiiiiiaioe eio?eo ii?iaeaeothoue ia a?eueio
a?oio, ia? i??o 0. Oae, Ae.Aiooaei aeia?a, ui a?aeiinii i??e Oaa?a
«a?eueo?noue» ne?i/aiii ii?iaeaeaieo i?aea?oi ca’ycii?,
ne?i/aiiiaei??ii? ia?ica’ycii? a?oie E? a?euei?.

, aeiiiaiaiiy oe??? iiiaeeie ? u?eueiei ? oiio iiaeia aiai?eoe i?i
«anthaeenou?noue» a?eueieo a?oi na?aae ne?i/aiii ii?iaeaeaieo i?aea?oi
aeaii? a?oie.

) ? a?eueieie. A?eueoa oiai, a oe?e aea ?iaio? aeiaaaeaii, ui a?euei?
i?aea?oie ? anthaeenoueie ? na?aae aeniei o?aiceoeaieo i?aea?oi
neiao?e/ii? a?oie iane?i/aiiiai noaiaiy.

Ia?aoo?, I.Aoaoa/a?aee iieacaea, ui «a?eueo?noue» ne?i/aiii ii?iaeaeaieo
i?aea?oi a?ioeaaeo aeiaooe?a ca iane?i/aiieie iine?aeiaiinoyie
iao?ea?aeueieo a?oi ? a?eueieie ye o eaoaai?iiio ?icoi?ii?, oae ? a
?icoi?ii? i??e.

A aeena?oaoe?ei?e ?iaio? aea/athoueny cia?aaeaiiy a?eueieo a?oi oa
iai?aa?oi aaoiiaoieie i?aenoaiiaeaie iaae ne?i/aiiei aeoaa?oii. I?e
oeueiio aaoiiaoieie ie iaceaa?ii ia?aoai?aiiy iiiaeeie iane?i/aiieo ne?a
iaae aeaiei aeoaa?oii, ye? iiaeooue aooe caaeai? aeayeei (ne?i/aiiei aai
iane?i/aiiei) aaoiiaoii. Aeia?a a?aeiii, ui aaoiiaoieie ? o? ? eeoa o?
ia?aoai?aiiy, eio?? ia?e ne?a c ??aieie ii/aoeaie iaaii? aeiaaeeie
ia?aaiaeyoue o neiaa, ii/aoee yeeo o??? ae aeiaaeeie oaeiae ??ai?.

A?oie aaoiiaoieo i?aenoaiiaie ? ia’?eoii /eneaiieo aeine?aeaeaiue
inoaii?o ?ie?a. Aeei?enoiaoth/e ca’ycie oeeo a?oi c a?ioeaaeie
aeiaooeaie ca iane?i/aiieie iine?aeiaiinoyie a?oi i?aenoaiiaie oa
a?oiaie aaoiii?o?ci?a ?aaoey?ieo ei?aiaaeo aea?aa, ??cieie aaoi?aie
aoaeoaaeenue i?eeeaaee iane?i/aiieo ne?i/aiii ii?iaeaeaieo ia??iaee/ieo
a?oi, i?eeeaaee a?oi i?ii?aeiiai ?inoo oa aiaiaaaeuei? iaaeaiaioa?i?
a?oie ye i?aea?oie a?oie aaoiiaoieo i?aenoaiiaie iaae aeayeei aeoaa?oii.
?acii c oei, aeine?aeaeaiue aoaeiae naieo a?oi aaoiiaoieo i?aenoaiiaie
aoei iaei, a aoaeiaa

iai?aa?oie aaoiiaoieo ia?aoai?aiue ia aea/aeanue acaaae?.

Iaoa ?iaioe. Iaoith aeenapoaoe?eii? piaioe ?

?icaeiooe oaoi?/iee aia?ao ia/eneaiue a iai?aa?oiao aaoiiaoieo
ia?aoai?aiue oa a?oiao ne?i/aiii aaoiiaoieo i?aenoaiiaie ? iienaoe
a?aeiioaiiy A??ia a iai?aa?oi? an?o aaoiiaoieo ia?aoai?aiue iaae
ne?i/aiiei aeoaa?oii;

) i?aeiai?aa?oi oe??? iai?aa?oie ? a?eueieie ? iiaoaeoaaoe eiie?aoi?
i?eeeaaee a?eueieo iai?aa?oi aaoiiaoieo ia?aoai?aiue;

ciaeoe yai? cia?aaeaiiy a?eueii? a?oie ?aiao 2 aaoiiaoieie
i?aenoaiiaeaie iaae aeaiaeaiaioiei aeoaa?oii;

ne?i/aiii aaoiiaoieie i?aenoaiiaeaie iaae o?eaeaiaioiei aeoaa?oii.

Iaoiaee aeine?aeaeaiiy. O aeenapoaoe?ei?e piaio? aeeipenoiaothoueny
iaoiaee oaip?? apoi oa iai?aa?oi, ciepaia eiia?iaoi?ii? oaip?? apoi oa
iai?aa?oi, a oaeiae oiiieia?/i? iaoiaee.

Iaoeiaa iiaecia. A aeena?oaoe?ei?e ?iaio? io?eiaii oae? iia?
?acoeueoaoe:

aeiaaaeaii, ui a?eueo?noue (o eaoaai?iiio ?icoi?ii? Aa?a) ne?i/aiii
ii?iaeaeaieo i?aeiai?aa?oi iai?aa?oie aaoiiaoieo ia?aoai?aiue iaae
ne?i/aiiei aeoaa?oii ? a?eueieie;

a oe?e iai?aa?oi? iienaii a?aeiioaiiy A??ia;

ciaeaeaii caaaeuei? eiino?oeoe?? a?eueieo iai?aa?oi aaoiiaoieo
ia?aoai?aiue ? iiaoaeiaaii eiioeioaeueio ?iaeeio oaeeo iai?aa?oi;

iaaaaeaii ?yae cia?aaeaiue iaaaaeaai? a?eueii? a?oie aaoiiaoieie
i?aenoaiiaeaie iaae aeaiaeaiaioiei aeoaa?oii, eio?? ia ? ne?i/aiii
aaoiiaoieie;

aeacaii cia?aaeaiiy a?eueii? a?oie ?aiao 2 ne?i/aiii aaoiiaoieie
i?aenoaiiaeaie iaae aeoaa?oii ?c aeaio aeaiaio?a.

Oai?aoe/ia ? i?aeoe/ia oe?ii?noue aeena?oaoe??. ?acoeueoaoe
aeenapoaoe?? ? iaaiei aianeii a eiia?iaoi?io oai??th apoi oa iai?aa?oi.
Iiaoaeiaai? i?eeeaaee oa oaoi?ea ia/eneaiue iiaeooue aooe aeeipenoai?
aeey iiaeaeueoeo aeine?aeaeaiue iai?aa?oi aaoiiaoieo ia?aoai?aiue oa
a?oi aaoiiaoieo i?aenoaiiaie, a oaeiae a?eueieo a?oi oa iai?aa?oi.

Ai?iaaoe?y ?iaioe. ?acoeueoaoe, iopeiai? a aeenapoaoe??,
aeiiia?aeaeeny: ia nai?iap? «Oaip?y apoi oa iai?aapoi» o Ee?anueeiio
Oi?aapneoao? ?iai? Oapana Oaa/aiea; ia Ee?anueeiio aeaaa?a?/iiio
nai?ia??; ia Anaoe?a?inuee?e iaoeia?e eiioa?aioe?? «?ic?iaea oa
canoinoaaiiy iaoaiaoe/ieo iaoiae?a a iaoeiai-oaoi?/ieo aeine?aeaeaiiyo»
(Euea?a, 1995); ia I’yo?e I?aeiapiaei?e eiioapaioe?? ?iai? aeaaeai?ea
I.Epaa/oea (Ee?a, 1996), a oaeiae ia i?aeiapiaei?e eiioapaioe?? «Groups
and Group Rings VI» (Wisla, 1998).

Ioae?eaoe??. Iniiai? pacoeueoaoe aeenapoaoe?? iioae?eiaaii a 4 iaoeiaeo
noaooyo, a oaeiae a 3 oacao aeiiia?aeae iaoeiaeo eiioa?aioe?e, nienie
yeeo iiaeaii a e?ioe? aaoi?aoa?aoo.

Iniaenoee aianie aeena?oaioa. An? pacoeueoaoe aeenapoaoe?? iopeiai?
aaoipii naiino?eii.

No?oeoo?a ? ia’?i ?iaioe. Aeenapoaoe?eia piaioa neeaaea?oueny c? anooio,
opueio picae?e?a, aeniiae?a ? nieneo e?oapaoope, aeeeaaeaieo ia 111
noip?ieao iaoeiiieniiai oaenoo. Nienie e?oapaoope i?noeoue 38
iaeiaioaaiue.

CI?NO ?IAIOE

O anooi? c?iaeaii ei?ioeee iaeyae e?oa?aoo?e ca oaiith aeena?oaoe??,
aeeeaaeaii iniiai? ?acoeueoaoe ?iaioe.

Ia?oee ?icae?e i?enay/aii icia/aiiyi oa iniiaiei aeanoeainoyi iai?aa?oi
aaoiiaoieo ia?aoai?aiue oa a?oi aaoiiaoieo i?aenoaiiaie. Ia?aa?aoe
1.1(1.3 iathoue aeiiii?aeiee oa?aeoa?.

, a ye?e:

Q — iaii?iaeiy, ia a?eueo i?ae ce?/aiia iiiaeeia, aeaiaioe yei?
iaceaathoueny aioo??oi?ie noaiaie aaoiiaoa;

— aeayeee aeae?eaiee aioo??oi?e noai aaoiiaoa, ?i?oe?aeueiee noai;

— ooieoe?y ia?aoiae?a;

— ooieoe?y aeoiae?a.

).

A ia?aa?ao? 1.2 aaiaeyoueny aei ?icaeyaeo iniiai? ia’?eoe aeine?aeaeaiiy
a

aeena?oaoe?ei?e ?iaio?. Oea, ianaiia?aae, iai?aa?oia AT(() aaoiiaoieo
ia?aoai?aiue iaae aeoaa?oii ( (a?aeiinii noia?iiceoe??). A oe?e
iai?aa?oi? aeae?eythoueny i?aeiai?aa?oia FAT(() ne?i/aiii aaoiiaoieo
ia?aoai?aiue, a oaeiae i?aea?oie AP(() aaoiiaoieo ? FAP(() ne?i/aiii
aaoiiaoieo i?aenoaiiaie (a??eoeaieo ia?aoai?aiue) iaae (. Cacia/a?oueny,
ui eiaeia c oeeo iai?aa?oi /e a?oi ae?? ia iiiaeeiao ne?i/aiieo ?
iane?i/aiieo ne?a iaae aeoaa?oii (.

anio oeo ia?aoai?aiue s iiiaeeie

,

o yeeo aeey aeiaieueii? iineiaeiaiinoi

cia/aiiy k(oi? eii?aeeiaoe ?? ia?aco

. Aea/athoueny iaei?ino?o? aeanoeaino? oe??? eiino?oeoe??. Anoaiiaeaii
oae? ?acoeueoaoe.

, ?ciii?oiee ye iai?aa?oia ia?aoai?aiue a?aie/i?e iai?aa?oi? cai?ioiiai
niaeo?o iine?aeiaiino? a?ioeaaeo aeiaooe?a

.

Aiiiii?o?ciaie, ye? aecia/athoue oaeee cai?ioiee niaeo?, ? i?e?iaei?
i?iaeooaaiiy.

Aeey aeia?eueiiai iaoo?aeueiiai k a?ioeaa? aeiaooee

iiae?ai? ye iai?aa?oie ia?aoai?aiue.

sseui k-oee iiiaeiee a?ioeaaiai aeiaooeo iaea?aeaii ye ia?aoei aeayei?
?iaeeie iai?aa?oi, oi nai a?ioeaaee aeiaooie ? ia?aoeiii a?ioeaaeo
aeiaooe?a, o yeeo k-o? iiiaeieee i?ia?aathoue oeth ?iaeeio iai?aa?oi, a
an? ?io? ca?aathoueny c iiiaeieeaie ii/aoeiaiai a?ioeaaiai aeiaooeo.

sseui k-oee iiiaeiee a?ioeaaiai aeiaooeo ii?iaeaeo?oueny aeayeith
?iaeeiith iai?aa?oi, ia?o? k-1 iiiaeiee?a ae?thoue ia ne?i/aiieo
iiiaeeiao ? an? iai?aa?oie i?noyoue iaeeieoeth, oi nai a?ioeaaee
aeiaooie ii?iaeaeo?oueny a?ioeaaeie aeiaooeaie, o yeeo k-o? iiiaeieee
i?ia?aathoue oeth ae naio ?iaeeio iai?aa?oi, a an? ?io? ca?aathoueny c
iiiaeieeaie ii/aoeiaiai a?ioeaaiai aeiaooeo.

, o?aiceoeaii ae?? ia iiiaeei? M oiae? e eeoa oiae?, eiee eiaeai c
iiiaeiee?a ? o?aiceoeaiith iai?aa?oiith.

Aeiaiaeeoueny

noaiaiy n=(((:

),

.

Ca?aene iaea?aeaii

(n>1).

? iiai? i?iia?ace ne?a ?c ranF. Iniiaiee ?acoeueoao oeueiai ia?aa?aoa
oi?ioeth?oueny oaeei /eiii.

AT((). Oiae?

1. F L G ( ranF=ranG;

;

;

AP((), ui ran(FH)=ranG;

5. J = D.

Caoaaaeeii, ui oey oai?aia ? i?e?iaeiei ocaaaeueiaiiyi iieno a?aeiioaiue
A??ia a neiao?e/i?e iai?aa?oi?.

Ae?oaee ?icae?e i?enay/aii a?eueiei i?aeiai?aa?oiai iai?aa?oie
aaoiiaoieo ia?aoai?aiue iaae ne?i/aiiei aeoaa?oii.

) aeaea?o?a noai?iue oeueiai i?inoi?o.

, eiiiiiaioe yeeo ii?iaeaeothoue a?eueio iai?aa?oio ?aiao k. Iniiaieie
?acoeueoaoaie ia?aa?aoa ? oae? aea? oai?aie, ye? aeicaieythoue
noaa?aeaeoaaoe, ui a eaoaai?iiio ?icoi?ii? «a?eueo?noue» k-ii?iaeaeaieo
i?aeiai?aa?oi iai?aa?oie AT(() ? a?eueieie iai?aa?oiaie ?aiao k.

.

.

? ?icaeyaea?oueny i?aea?oia oeo aaoiiaoieo i?aenoaiiaie, ye? iiaeia
caaeaaaoe ca aeiiiiiaith iane?i/aiieo oaaeeoeue

oaeeo, ui

,

, ii?iaeaeaiee iiiai/eaiaie

.

?icaeyaea?oueny aeia?eueia iine?aeiai?noue

.

Oai?aia 2.9 anoaiiaeth?, ui i?e p>2 oaaeeoe?

.

A oai?ai? 2.10 aeiaaaeaii, ui oaaeeoeyie

.

Oe? aea? oai?aie aeicaieythoue noi?ioethaaoe

(p — i?inoa) i?noeoue eiioeiooi aaaaoi iiia?ii ??cieo a?eueieo
i?aeiai?aa?oi ?aiao 2.

Aea/aii ieoaiiy ne?i/aiii? aaoiiaoiino? iiaoaeiaaieo iai?aa?oi.
Anoaiiaethoueny (oaa?aeaeaiiy 2.13), ui oe? iai?aa?oie aoaeooue
ne?i/aiii aaoiiaoieie a oiio ? eeoa oiio ?ac?, eiee iine?aeiai?noue a ?
iaeaea ia??iaee/iith, oiaoi ia??iaee/iith, ii/eiath/e c aeayeiai i?noey.

.

.

O ia?aa?ao? 3.1 aoaeo?oueny na??y 2-ii?iaeaeaieo i?aea?oi oe??? a?oie.
Aeiaiaeeoueny (oai?aie 3.2, 3.3), ui eiaeia ?c iiaoaeiaaieo a?oi ?
a?eueiith a?oiith ?aiao 2. O oaa?aeaeaii? 3.4 anoaiiaeth?oueny, ui an?
aiie ia ? a?oiaie ne?i/aiii aaoiiaoieo i?aenoaiiaie.

Iniiaiei ?acoeueoaoii ia?aa?aoa 3.2 ?

ae?oaiai ii?yaeeo.

iathoue ii?yaeie 2 ? oaeeo, ui a?oia, ieie ii?iaeaeaia,
?iceeaaea?oueny o a?eueiee aeiaooie k nai?o oeeee?/ieo i?aea?oi ae?oaiai
ii?yaeeo.

. E?euee?noue aioo??oi?o noai?a oeeo aaoiiao?a ??aia a?aeiia?aeii 10,
10 ? 8. Aaoiiaoi? i?aenoaiiaee, aecia/ai? oeeie aaoiiaoaie, caaeathoueny
oaeeie oaaeeoeyie:

Iaaiaeeoueny eiino?oeoeaia aeiaaaeaiiy ?nioaaiiy o a?oi? ne?i/aiii
aaoiiaoieo i?aenoaiiaie iaae aeaiaeaiaioiei aeoaa?oii i?aea?oie,
?ciii?oii? a?eueiiio aeiaooeo oeeee?/ii? a?oie ae?oaiai ii?yaeeo ?
iane?i/aiii? oeeee?/ii? a?oie.

?icaeyaeathoueny oaaeeoe?

? aeiaiaeeoueny

a?eueii ii?iaeaeothoue a?eueio i?aea?oio ?aiao 2 o a?oi? ne?i/aiii
aaoiiaoieo i?aenoaiiaie iaae aeaiaeaiaioiei aeoaa?oii.

Cacia/eii, ui oea ? ia?oee i?eeeaae a?eueii? iaaaaeaai? a?oie ne?i/aiii
aaoiiaoieo i?aenoaiiaie iaae aeaiaeaiaioiei aeoaa?oii, yeee
noi?iaiaeaeo?oueny iiaiei aeiaaaeaiiyi. O ?iaio? N.A.Aeueioeia oaeiae
aoaeoaaany i?eeeaae oaei? a?oie, i?ioa iiaia aeiaaaeaiiy aaoi?ii aein?
ia iioae?eiaaia, a ni?iae ?ioeo iaoaiaoee?a caaa?oeoe eiai ia aeaee
aeiaeieo ?acoeueoao?a.

Aeae? iaaiaeeoueny

.

Ca aeiiiiiaith oe??? eaie, ei?enooth/enue oai?aiith 3.9, ie aeiaiaeeii

ne?i/aiii aaoiiaoieo i?aenoaiiaie iaae o?eaeaiaioiei aeoaa?oii.

aeaiaio?a o?aoueiai ii?yaeeo iaia?.

ne?i/aiii aaoiiaoieo i?aenoaiiaie iaae aeaiaeaiaioiei aeoaa?oii,
oa??i? yei? caaeathoueny aaoiiaoaie, cia?aaeaieie ia ?enoieo 1 (o
aeena?oaoe?ei?e ?iaio? oeueiio ?enoieo a?aeiia?aeathoue ?enoiee 3.4 ?
3.5).

?en.1

Aecia/ai? oeeie aaoiiaoaie aaoiiaoi? i?aenoaiiaee caaeathoueny oaeeie
oaaeeoeyie a?aeiia?aeii:

?

.

AENIIAEE

A aeena?oaoe?ei?e ?iaio? aea/athoueny a?euei? i?aea?oie oa
i?aeiai?aa?oie iai?aa?oie aaoiiaoieo ia?aoai?aiue iaae ne?i/aiiei
aeoaa?oii.

Iieacaii, ui a?eueieo i?aeiai?aa?oi «a?eueo?noue» (o ?icoi?ii?
eaoaai??? Aa?a) na?aae ne?i/aiii ii?iaeaeaieo i?aeiai?aa?oi oe???
iai?aa?oie. Oaeiae ?icaeiooi oaoi?eo ia/eneaiue o a?ioeaaeo aeiaooeao
iai?aa?oi ia?aoai?aiue oa a?oi i?aenoaiiaie, caaaeyee /iio iiaoaeiaaii
eiioeioaeueio ?iaeeio a?eueieo i?aeiai?aa?oi iai?aa?oie aaoiiaoieo
ia?aoai?aiue, aeacaii na??th a?eueieo i?aea?oi oe??? iai?aa?oie.
Ei?enooth/enue iiaaie ne?i/aiieo aaoiiao?a oa a?ioeaaeo aeiaooe?a
iai?aa?oi ca iane?i/aiieie iine?aeiaiinoyie iai?aa?oi ia?aoai?aiue,
iiaoaeiaaii ne?i/aiii aaoiiaoi? iaae aeaiaeaiaioiei aeoaa?oii a?oie,
?ciii?oi? a?eueiei aeiaooeai ne?i/aiii? e?eueeino? oeeee?/ieo a?oi
ae?oaiai ii?yaeeo, a oaeiae a?euei?e a?oi? ?aiao 2.

Canoiniaai? a aeena?oaoe?ei?e ?iaio? iaoiaee ? iiaoaeiaai? i?eeeaaee
iiaeia aoaea aeei?enoiaoaaoe aeey iiaeaeueoiai aeine?aeaeaiiy iai?aa?oi
aaoiiaoieo ia?aoai?aiue ? a?oi aaoiiaoieo i?aenoaiiaie, a oaeiae
a?eueieo a?oi oa iai?aa?oi.

Aaoi? aeneiaeth? ue?o iiaeyeo nai?io iaoeiaiio ea??aieeo i?ioani?o
Nouainueeiio A?oae?th ?aaiiae/o ca iino?eio oaaao ? i?aeo?eieo a ?iaio?.

?IAIOE AAOIPA CA OAIITH AeENAPOAOe??

Ieieiee A.C. Aieueii iaiiaa?oie neii/aiii aaoiiaoieo ia?aoai?aiue//
Anaoe?a?inueea iaoeiaa eiioa?aioeiy «?ic?iaea oa canoinoaaiiy
iaoaiaoe/ieo iaoiaeia a iaoeiai-oaoii/ieo aeineiaeaeaiiyo».— Eueaia.—
1995.— Oace aeiiia?aeae. n.39.

Ieieiee A.C. Cia?aaeaiiy a?eueieo iai?aa?oi eaac?ae?aaiiaeueieie
aaoiiaoieie i?aenoaiiaeaie// I’yoa I?aeiapiaeia eiioapaioe?y ?iai?
aeaaeai?ea I.Epaa/oea.— Ee?a.— 1996.— Oace aeiiia?aeae. n.307.

Ie?eiee A. A?aeiioaiiy A??ia a iai?aa?oiao aaoiiaoieo ia?aoai?aiue//
A?niee Ee?anueeiai oi-oo. Na??y o?c.-iao.— 1997.— aei.4.— n.91(97.

Ieeeiue A.N. I naiaiaeiuo iieoa?oiiao aaoiiaoiuo i?aia?aciaaiee// Iaoai.
caiaoee.— 1998.— 63.— n.248(259.

Ie?eiee A.N. A?euei? a?oie aaoiiaoieo i?aenoaiiaie// Aeii. IAI Oe?a?ie.—
1998.— ? 7.— n.40(44.

and finite automata// Abstr. Int. Conf. Group and Group Rings VI.—
Wisla.— 1998.— p.28.

as groups of finitely automatic permutations// Aii?inu aeaaa?u.—
1999.— aui.14.— n.158(165.

Ie?eiee A.N. A?euei? a?oie oa iai?aa?oie aaoiiaoieo ia?aoai?aiue. —

?oeiien.

Aeenapoaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeeoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.06 — aeaaapa ? oaip?y
/enae. — Ee?anueeee oi?aapneoao ?iai? Oapana Oaa/aiea, i.Ee?a, 1999.

Aeena?oaoe?th i?enay/aii aea/aiith cia?aaeaiue a?eueieo a?oi oa
iai?aa?oi aaoiiaoieie ia?aoai?aiiyie iaae ne?i/aiieie aeoaa?oaie.
?icaeiooi oaoi?eo ia/eneaiue o a?ioeaaeo aeiaooeao ca iane?i/aiieie
iine?aeiaiinoyie iai?aa?oi ia?aoai?aiue. Aeiaaaeaii, ui o nain? Aa?a
«a?eueo?noue» ne?i/aiii ii?iaeaeaieo i?aeiai?aa?oi iai?aa?oie aaoiiaoieo
ia?aoai?aiue iaae ne?i/aiiei aeoaa?oii ? a?eueieie. Iiaoaeiaaii
eiioeioaeueio na??th i?eeeaae?a a?eueieo iai?aa?oi aaoiiaoieo
i?aenoaiiaie.

Ciaeaeaii cia?aaeaiiy a?eueieo iaaaaeaaeo a?oi ?aiao 2 ye ne?i/aiieie,
oae ? iane?i/aiieie aaoiiaoaie iaae aeaiaeaiaioiei aeoaa?oii.

Iienaii a?aeiioaiiy A??ia a iai?aa?oi? aaoiiaoieo ia?aoai?aiue iaae
ne?i/aiiei aeoaa?oii.

Eeth/ia? neiaa: aaoiiao, ne?i/aiiee aaoiiao, aaoiiaoia ia?aoai?aiiy,
aaoiiaoia i?aenoaiiaea, a?ioeaaee aeiaooie, a?eueia iai?aa?oia, a?eueia
a?oia, a?aeiioaiiy A??ia.

Ieeeiue A.N. Naiaiaeiua a?oiiu e iieoa?oiiu aaoiiaoiuo i?aia?aciaaiee. —
?oeiienue.

Aeennapoaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.06 — aeaaapa e
oaipey /enae. — Eeaaneee oieaapneoao eiaie Oapana Oaa/aiei, a.Eeaa,
1999.

Aeenna?oaoeey iinayuaia eco/aieth i?aaenoaaeaiee naiaiaeiuo
a?oii e iieoa?oii aaoiiaoiuie i?aia?aciaaieyie iaae eiia/iuie
aeoaaeoaie. ?icaeoa oaoieea au/eneaiee a nieaoaieyo ii aaneiia/iui
iineaaeiaaoaeueiinoyi iieoa?oii i?aia?aciaaiee. Aeieacaii, /oi a niunea
Ay?a «aieueoeinoai» eiia/ii ii?iaeaeaiiuo iiaeiieoa?oii iieoa?oiiu
aaoiiaoiuo i?aia?aciaaiee iaae eiia/iui aeoaaeoii yaeythony naiaiaeiuie.
Iino?iaia eiioeioaeueiay na?ey i?eia?ia naiaiaeiuo iieoa?oii aaoiiaoieo
iiaenoaiiaie.

Iaeaeaii i?aaenoaaeaiey naiaiaeiuo iaaaaeaauo a?oii ?aiaa 2 eae
eiia/iuie, oae e aaneiia/iuie aaoiiaoaie iaae aeaooyeaiaioiui aeoaaeoii.

Iienaiu ioiioaiey A?eia a iieoa?oiia aaoiiaoiuo i?aia?aciaaiee iaae
eiia/iui aeoaaeoii.

Eeth/aaua neiaa: aaoiiao, eiia/iue aaoiiao, aaoiiaoiia i?aia?aciaaiea,
aaoiiaoiay iiaenoaiiaea, nieaoaiea, naiaiaeiay iieoa?oiia, naiaiaeiay
a?oiia, ioiioaiey A?eia.

Olijnyk A.S. Free groups and semigroups of automaton transformations. —
Manuscript.

Thesis of a dissertation for obtaining the degree of candidate of
sciences

in physics and mathematics, speciality 01.01.06 — algebra and number
theory. — Kyiv Taras Shevchenko University, Kyiv, 1999.

Semigroup of automaton transformations over finite alphabets are
considered in the dissertation. A notion of the wreath product over an
infinite sequence of transformation semigroups is introdused. Basic
properties of this construction are described. It is pointed, that the
semigroup of automaton transformation over finite alphabet is similar
as a transformation semigroup to the wreath product over a sequence of
symmetric semigroups over this alphabet.

Green relations in the semigroup of automaton transformations over the
finite alphabet are described. This description is a nature
generalization of that in symmetric semigroups.

Representations of free groups and free semigroups by automaton
transformations over finite alphabets are investigated. A technique of
calculations in wreath products over infinite sequences of semigroups of
transformations is developed. It is proved that «most» (due to Baire)
finitely generated subsemigroups of the semigroup of automaton
transformation over the finite alphabet are free. More precisely, we
consider a metric on this semigroup. Obtained metric space and
arbitrary its direct product are complete. The semigroup operation is
continuous under this metric. The set of all k- tuples which components
generates a free semigroup is dense and its complement is a meagre set
(or a set of the first category) in the corresponding metric space.
Note, that similar results about free subgroups in certain groups were
obtained in papers of J.D.Dixon and M.Bhattacharjee. Theorems about the
ubiquity (in different meanings) of free subgroups in some groups are
started to prove since the famuous Tits alternative was formulated.

A continuum serie of free semigroups of automaton permutations over
alphabets with prime number elements is constructed. Finitely
automaton ones are distinguished.

Representations of the free non-abelian group of rank 2 by infinite
automata over two-element alphabet are pointed.

It is proved that a free product of arbitrary finite number of cyclic
groups of order 2 is isomorphic to some subgroup of the group of
finitely automaton permutations over alphabet identifying with the prime
field of characteristic 2. A proof is constructive. For example, if we
consider a free product of three cyclic groups of order 2 then
generating automata have 10, 10 and 8 inner states correspondingly.

A construction of free finitely automaton group over a two-element
alphabet is presented.

It is pointed that the general linear group of degree 2 over integers is
a subgroup of the group of finitely automaton permutations over a
3-element alphabet. It gives a positive answer to the question of
Brunner-Sidki.

Automata with 4 inner states over the alphabet(0,1( generating free
abelian group of rank 2 are presented.

Key words: automaton, finite automaton, automaton transformation,
automaton permutation, wreath product, free semigroup, free group,
Green relations.

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