Congruences of a Permutable Inverse Semigroup of Finite Rank
We describe the structure of any congruence of a permutable inverse semigroup of finite rank.
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| Sprache: | Ukrainisch Englisch |
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509732593205248 |
|---|---|
| author | Derech, V. D. Дереч, В. Д. |
| author_facet | Derech, V. D. Дереч, В. Д. |
| author_sort | Derech, V. D. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:05Z |
| description | We describe the structure of any congruence of a permutable inverse semigroup of finite rank. |
| first_indexed | 2026-03-24T02:45:47Z |
| format | Article |
| fulltext |
UDK 512.534.5
V. D. Dereç (Vinnyc. nac. texn. un-t)
KONHRUENCI} PERESTAVNO} INVERSNO}
NAPIVHRUPY SKINÇENNOHO RANHU
The structure of any congruence of a permutable inverse semigroup of a finite rank is described.
Opysano budovu bud\-qko] konhruenci] perestavno] inversno] napivhrupy skinçennoho ranhu.
Qk vidomo, bud\-qki dvi konhruenci] na hrupi perestavni vidnosno zvyçajno] ope-
raci] superpozyci] binarnyx vidnoßen\. Oçevydno, wo vidpovidna vlastyvist\ ma[
misce i dlq tyx alhebra]çnyx struktur, do skladu qkyx vxodyt\ hrupova struk-
tura (kil\cq, moduli ta inße). Do klasu binarnyx alhebr z perestavnymy kon-
hruenciqmy takoΩ naleΩat\ ekvazihrupy i skinçenni kvazihrupy. Wo stosu[t\sq
teori] napivhrup, to tut vidomi xiba wo klasy napivhrup z perestavnymy konhru-
enciqmy (napivhrupy Brandta, vsi vydy skinçennyx symetryçnyx napivhrup, na-
pivhrupy endomorfizmiv, a takoΩ çastkovyx avtomorfizmiv skinçenno] linijno
vporqdkovano] mnoΩyny ta inßi). U statti [1] znajdeno neobxidni i dostatni
umovy dlq toho, wob antyhrupa skinçennoho ranhu bula perestavnog (oznaçennq
dyv. v p.41). Dana stattq [ sutt[vym uzahal\nennqm rezul\tatu roboty [1]. U nij
(p.44, teorema44) z’qsovu[t\sq budova bud\-qko] konhruenci] perestavno] inversno]
napivhrupy z nulem skinçennoho ranhu.
1. Osnovna terminolohiq i poznaçennq. Napivreßitku S nazyvagt\ napiv-
reßitkog skinçenno] dovΩyny, qkwo isnu[ natural\ne çyslo n take, wo dov-
Ωyna bud\-qkoho lancgΩka z S ne perevywu[ n.
Nexaj P — vporqdkovana mnoΩyna skinçenno] dovΩyny z najmenßym ele-
mentom 0. Toçna verxnq meΩa dovΩyn lancgΩkiv, wo z’[dnugt\ 0 i element
x, nazyva[t\sq vysotog elementa x i poznaça[t\sq çerez h ( x ) .
Nexaj S — dovil\na napivhrupa, a N0 — mnoΩyna vsix nevid’[mnyx cilyx çy-
sel. Funkcig rank : S → N0 nazyvagt\ ranhovog na napivhrupi S, qkwo dlq
bud\-qkyx elementiv a i b ∈ S vykonu[t\sq nerivnist\
rank ( a ⋅ b ) ≤ min { rank ( a ) , rank ( b ) } .
Çyslo rank ( a ) nazyva[t\sq ranhom elementa a.
Napivhrupu nazyvagt\ perestavnog, qkwo bud\-qki dvi konhruenci] na nij
perestavni vidnosno zvyçajno] operaci] superpozyci] binarnyx vidnoßen\.
Vsi inßi neobxidni ponqttq z teori] napivhrup i teori] inversnyx napivhrup
moΩna znajty vidpovidno v monohrafiqx [2, 3].
2. Ranhova funkciq i ]] osnovni vlastyvosti. V c\omu punkti vvedemo ran-
hovu funkcig na inversnij napivhrupi, napivreßitka idempotentiv qko] ma[ skin-
çennu dovΩynu, i vstanovymo osnovni ]] vlastyvosti. Dlq c\oho spoçatku damo
vyznaçennq ranhu elementa napivreßitky skinçenno] dovΩyny.
OtΩe, nexaj P — napivreßitka skinçenno] dovΩyny. Za oznaçennqm
rank ( a ) = h ( a ) (de h ( a ) — vysota elementa a ).
Lema.1. Wojno vyznaçena funkciq rank : P → N0 [ ranhovog.
Ce lehko pereviryty.
Lema.2. Qkwo a < b , to rank ( a ) < rank ( b ) .
Dovedennq. Lehko perevirq[t\sq.
Dali, nexaj S — inversna napivhrupa, napivreßitka idempotentiv qko] ma[
skinçennu dovΩynu.
Oznaçennq. Nexaj a — dovil\nyj element napivhrupy S , todi (za ozna-
çennqm) rank ( a ) = rank ( a ⋅ a
–
1
) .
© V. D. DEREÇ, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4 469
470 V. D. DEREÇ
Teper nam potribno dovesty, wo rank : S → N0 dijsno [ ranhovog funkci[g
na inversnij napivhrupi S.
Dlq c\oho dovedemo kil\ka lem.
Lema.3. Nexaj A i B — napivreßitky skinçenno] dovΩyny, f : A → B —
sgr’[ktyvnyj homomorfizm. Todi dlq bud\-qkoho b ∈ B pidnapivreßitka
f –
1 ( b ) ma[ najmenßyj element.
Dovedennq. Nexaj a1 < a2 < … < ak — maksymal\nyj (oçevydno, wo takyj
isnu[) lancgΩok, qkyj vklgça[t\sq v f b−1( ). Todi a1 — minimal\nyj element
v f b−1( ), a otΩe, a1 — najmenßyj element v f b−1( ).
Lema.4. Nexaj A i B — napivreßitky skinçenno] dovΩyny, f : A → B —
sgr’[ktyvnyj homomorfizm. Qkwo b b1 2< (de b B1 ∈ i b B2 ∈ ), to isnugt\
a A1 ∈ i a A2 ∈ taki, wo a a1 2< i f a b( )1 1= , f a b( )2 2= .
Dovedennq. Za poperedn\og lemog v pidnapivreßitci f b−1
1( ) [ najmen-
ßyj element. Poznaçymo joho çerez a1. Nexaj a2 — dovil\nyj element z
mnoΩyny f b−1
2( ) . Oskil\ky f a a( )1 2⋅ = f a f a( ) ( )1 2⋅ = b b1 2⋅ = b1, to a a1 2⋅ ∈
∈ f b−1
1( ). Element a1 [ najmenßym u f b−1
1( ), tomu a1 ≤ a a1 2⋅ . Z ostann\o]
nerivnosti lehko vyplyva[ a1 ≤ a2 . Oskil\ky a1 ≠ a2 , to a1 < a2 .
Lema.5. Nexaj A i B — napivreßitky skinçenno] dovΩyny, f : A → B —
sgr’[ktyvnyj homomorfizm. Todi dlq bud\-qkoho a ∈ A vykonu[t\sq nerivnist\
rank ( f ( a )) ≤ rank ( a ) .
Dovedennq. Oskil\ky B — napivreßitka skinçenno] dovΩyny, to v nij [
najmenßyj element 0. Nexaj 0 < b1 < b 2 < … < bn – 1 < f ( a ) = bn —
maksymal\nyj (za kil\kistg elementiv) lancgΩok, wo z’[dnu[ 0 i f ( a ) . U pid-
napivreßitkax f −1 0( ), f b−1
1( ), f b−1
2( ), … , f bn
−
−
1
1( ) vybyra[mo najmenßi ele-
menty, vidpovidno 0, a1, a2, a3, … , an – 1 . Za lemog44 ce lancgΩok 0 < a1 <
< a2 < … < an – 1 , dovΩyna qkoho dorivng[ n – 1. Tomu lancgΩok 0 < a1 <
< a2 < … < an – 1 < a ma[ dovΩynu n. OtΩe, rank ( f ( a )) ≤ rank ( a ) .
Teorema.1. Nexaj S — inversna napivhrupa, napivreßitka idempotentiv
qko] ma[ skinçennu dovΩynu. Todi funkciq rank : S → N0 (vona vyznaçena v p. 2)
[ ranhovog.
Dovedennq. Nexaj a i b — dovil\ni elementy, wo naleΩat\ S. Spoçatku
dovedemo, wo rank ( a ⋅ b ) ≤ rank ( a ) . Dijsno, za oznaçennqm
rank ( )ab = rank(a bb a− −1 1) .
Oskil\ky abb a− −1 1 ≤ aa−1, to (za lemog42) rank ( )abb a− −1 1 ≤ rank ( )aa−1
.
OtΩe,
rank ( )ab = rank ( )abb a− −1 1 ≤ rank ( )aa−1 = rank ( a ) .
Teper dovedemo, wo rank ( )ab ≤ rank ( b ) .
Za oznaçennqm rank ( )ab = rank ( )abb a− −1 1 . Vidomo (i ce lehko pereviryty),
wo funkciq f e aeaa : � −1
[ endomorfizmom napivreßitky E — idempotentiv
napivhrupy S. Zokrema, f bb E abb a Ea : − − −1 1 1� — sgr’[ktyvnyj homomorfizm
z idealu bb E−1
na ideal abb a E− −1 1 , pryçomu f bba( )−1 = abb a− −1 1. Za le-
mog45 rank( )( )f bba
−1 ≤ rank ( )bb−1
abo rank ( )abb a− −1 1 ≤ rank ( )bb−1 . OtΩe,
rank ( )ab = rank ( )abb a− −1 1 ≤ rank ( )bb−1 = rank ( b ) . Takym çynom, funkciq
rank [ ranhovog.
Teper vidmitymo kil\ka osnovnyx vlastyvostej ranhovo] funkci].
Lema.6. Dlq bud\-qkoho a S∈ rank rank( ) ( )a a= −1
.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
KONHRUENCI} PERESTAVNO} INVERSNO} NAPIVHRUPY SKINÇENNOHO RANHU 471
Dovedennq. Ma[mo
rank ( a ) = rank ( )aa a−1 ≤ rank ( )aa−1 ≤ rank ( )a−1 .
Z inßoho boku,
rank ( )a−1 = rank ( )a aa− −1 1 ≤ rank ( )aa−1 ≤ rank ( a ) .
Takym çynom, rank ( a ) = rank ( )a−1 .
Lema.7. Dlq bud\-qkoho a S∈ rank ( a ) = rank ( )a a−1 .
Dovedennq. Spravdi, rank ( a ) = rank ( )a−1 = rank ( )a a−1 .
Lema.8. Qkwo a < b ( a ∈ S, b ∈ S ) , to rank ( a ) < rank ( b ) .
Dovedennq. PokaΩemo spoçatku, wo aa−1 < bb−1. Po-perße, zrozumilo,
wo aa−1 ≤ bb−1. Oskil\ky za umovog a < b , to a ba− −1 1 = a−1
. Zvidsy
a ba b− −1 1 = a b−1 , tobto a b−1
— idempotent. Prypustymo, wo aa−1 = bb−1,
todi b = aa b−1 . Oskil\ky a b−1
— idempotent, to b ≤ a. Supereçnist\. Takym
çynom, aa−1 < bb−1. Dali,
rank ( a ) = rank ( )aa−1 < rank ( )bb−1 = rank ( b ) .
OtΩe, rank ( a ) < rank ( b ) .
3. Neobxidna i dostatnq umova linijno] vporqdkovanosti idealiv. Vidomo
[4] (teorema44), wo idealy perestavno] napivhrupy utvorggt\ lancgΩok vidnos-
no vklgçennq. Oskil\ky dali mova jtyme pro perestavni inversni napivhrupy, to
dlq nas vaΩlyvo vstanovyty neobxidni i dostatni umovy dlq toho, wob idealy
tako] napivhrupy buly linijno vporqdkovanymy.
Spoçatku domovymosq pro terminolohig.
Nexaj S — inversna napivhrupa, napivreßitka idempotentiv qko] ma[ skinçen-
nu dovΩynu. Oçevydno, wo pidmnoΩyna Ik = { }( )x S x k∈ ≤rank napivhrupy S
[ idealom. Nazvemo takyj ideal ranhovym.
Budemo hovoryty, wo napivhrupa S zadovol\nq[ umovu L, qkwo dlq bud\-
qkyx a, b ∈ S z umovy rank ( a ) = rank ( b ) vyplyva[ SaS = SbS.
Teorema.2. Nexaj S — inversna napivhrupa, napivreßitka idempotentiv
qko] ma[ skinçennu dovΩynu. Nastupni umovy [ ekvivalentnymy:
1) idealy napivhrupy S linijno vporqdkovani;
2) holovni idealy napivhrupy S linijno vporqdkovani;
3) koΩnyj ideal napivhrupy S [ holovnym;
4) napivhrupa S zadovol\nq[ umovu L;
5) koΩnyj ideal napivhrupy S [ ranhovym.
Dovedennq. Implikaciq (1) → (2) oçevydna.
Implikaci] (2) → (1) i (3) → (2) magt\ misce dlq bud\-qko] napivhrupy i ce
lehko dovesty.
Dovedemo implikacig (1) → (4). OtΩe, nexaj a S∈ i b S∈ taki, wo
rank ( a ) = rank ( b ) = k. Potribno dovesty, wo SaS = SbS. Prypustymo proty-
leΩne, tobto SaS ≠ SbS. Idempotenty aa−1
i bb−1
poznaçymo vidpovidno
çerez e i w, a mnoΩynu { }x S SxS SeS∈ = — çerez De . Rozhlqnemo I Dk e−1 ∪ .
Dovedemo, wo I Dk e−1 ∪ [ idealom napivhrupy S.
Rozhlqnemo moΩlyvi vypadky.
1. Nexaj x Ik∈ −1. Todi dlq bud\-qkoho y S∈ ma[mo xy Ik∈ −1 i yx Ik∈ −1.
2. Nexaj teper x De∈ . Todi SxS = SeS. Nexaj c S∈ — dovil\nyj element
napivhrupy S. Qkwo rank ( xc ) < k – 1, to xc Ik∈ −1, a otΩe, xc I Dk e∈ −1 ∪ .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
472 V. D. DEREÇ
Nexaj teper rank ( xc ) = k . Todi rank ( )xcc x− −1 1 = k . Lehko pereviryty, wo
xcc x− −1 1 ≤ xx−1. Qkwo prypustyty, wo xcc x− −1 1 < xx−1, to za lemog48
rank ( )x cc x− −1 1 < rank ( )x x −1 . Oskil\ky rank ( )x cc x− −1 1 = k i rank ( )x x −1 = k,
to pryxodymo do supereçnosti. OtΩe, xcc x− −1 1 = xx−1.
Takym çynom,
SaS = Saa S−1 = SeS = SxS = Sxx S−1 = Sxcc x S− −1 1 = SxcS,
tobto xc De∈ .
Analohiçno moΩna dovesty, wo cx I Dk e∈ −1 ∪ .
Takym çynom, I Dk e−1 ∪ — ideal napihrupy S. Zrozumilo, wo I Dk w−1 ∪ —
teΩ ideal napivhrupy S.
Za umovog idealy napivhrupy S linijno vporqdkovani, otΩe,
I Dk e−1 ∪ ⊆ I Dk w−1 ∪ abo I Dk w−1 ∪ ⊆ I Dk e−1 ∪ .
A. Qkwo prypustyty, wo I Dk e−1 ∪ ⊆ I Dk w−1 ∪ , to e I Dk w∈ −1 ∪ . Oskil\-
ky rank ( e ) = k, to e Dw∈ . Zvidsy vyplyva[ SeS = SwS. Supereçnist\.
V. Qkwo Ω I Dk w−1 ∪ ⊆ I Dk e−1 ∪ , to w I Dk e∈ −1 ∪ . Zvidsy vyplyva[, wo
SeS = SwS. Supereçnist\.
Dovedemo spravedlyvist\ implikaci] (4) → (5).
OtΩe, nexaj napivhrupa S zadovol\nq[ umovu L. PokaΩemo, wo koΩnyj
ideal napivhrupy S [ ranhovym.
Nexaj I — dovil\nyj ideal. Poznaçymo çerez a element najbil\ßoho ranhu
sered usix elementiv idealu I. Nexaj rank ( a ) = k , a element b takyj, wo
rank ( b ) = k . Oskil\ky rank ( b ) = rank ( a ) , to SaS = SbS. Ale SaS ⊆ I, tomu
b ∈ I. Dali, ideal I mistyt\ idempotent ranhu k, a otΩe (ce lehko ob©runtu-
vaty), dlq bud\-qkoho nevid’[mnoho çysla m ( )m k< isnu[ idempotent, ranh qko-
ho dorivng[ m, pryçomu cej idempotent naleΩyt\ I. Mirkugçy tak samo, qk i
vywe, oderΩu[mo, wo vsi elementy ranhu m naleΩat\ idealu I. OtΩe, Ik = I.
Implikaciq (5) → (1) [ oçevydnog.
Dovedemo spravedlyvist\ implikaci] (5) → (3).
Nexaj I — dovil\nyj ideal napivhrupy S. Za umovog vin [ ranhovym, tobto
Ik = I dlq deqkoho k N∈ 0 . Dovedemo, wo vin [ holovnym. Nexaj a I∈ ,
pryçomu rank ( a ) = k . Rozhlqnemo holovnyj ideal SaS. Za umovog vin [
ranhovym. OtΩe, SaS = Im dlq deqkoho m N∈ 0 . Ale a SaS∈ , tomu rank ( a ) ≤
≤ m , tobto k ≤ m . Oskil\ky k ≤ m , to I Ik m⊆ . Krim c\oho, SaS I Ik⊆ = . Z
ostannix dvox vklgçen\ vyplyva[, wo I I SaSk m= = .
Dovedenyx implikacij dostatn\o, wob stverdΩuvaty poparnu ekvivalentnist\
umov (1) – (5).
4. Budova bud\-qko] konhruenci] perestavno] inversno] napivhrupy skin-
çennoho ranhu. V c\omu punkti dovedemo osnovnu teoremu statti (teoremu44).
Spoçatku sformulg[mo potribnyj rezul\tat (dyv. [1], p. 4).
Teorema.3. Nexaj S — napivhrupa, I1 i I 2 — ]] idealy, pryçomu I 1 ⊆ I2 .
Nexaj Θ1 ta Θ2 — konhruenci] na napivhrupi S taki, wo Θ1 = I I1 1× ∪ Ω i
Θ2 = I I2 2× ∪ σ , de Ω ⊆ H i σ ⊆ H, a H — vidnoßennq Hrina.
Todi Θ Θ1 2� = Θ Θ2 1� .
Teper sformulg[mo osnovnyj rezul\tat statti.
Teorema.4. Nexaj S — inversna napivhrupa z nulem 0, napivreßitka idem-
potentiv qko] ma[ skinçennu dovΩynu.
Bud\-qki dvi konhruenci] napivhrupy S perestavni todi i til\ky todi, koly ]]
idealy linijno vporqdkovani i koΩna konhruenciq Θ ma[ formu Θ = I I× ∪ Ω ,
de I — ideal napivhrupy S, Ω ⊆ H ( H — vidnoßennq Hrina).
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
KONHRUENCI} PERESTAVNO} INVERSNO} NAPIVHRUPY SKINÇENNOHO RANHU 473
Dovedennq. U teoremi43 v zahal\nij formi dovedeno dostatnist\. Dove-
demo teper neobxidnist\.
OtΩe, nexaj bud\-qki dvi konhruenci] na napivhrupi S [ perestavnymy. Todi,
qk vidomo [4] (teorema44), ]] idealy linijno vporqdkovani, a otΩe, za teoremog42
koΩnyj ideal napivhrupy S [ ranhovym. Dali, nexaj Θ — dovil\na konhruenciq
napivhrupy S. Lehko pereviryty, wo IΘ = { },x S x∈ 〈 〉 ∈0 Θ — ideal napivhru-
py S, otΩe, isnu[ çyslo k take, wo IΘ = Ik = { }( )x S x k∈ ≤rank .
Nexaj 〈 〉 ∈a b, Θ, pryçomu rank ( a ) > k . PokaΩemo, wo i rank ( b ) > k . Dijs-
no, qkwo prypustyty protyleΩne, tobto rank ( b ) ≤ k , to b Ik∈ , a tomu i
a Ik∈ . OtΩe, rank ( a ) ≤ k . Supereçnist\.
Teper pokaΩemo, wo rank ( a ) = rank ( b ) . Prypustymo protyleΩne, tobto
rank ( a ) ≠ rank ( b ) . Dlq konkretnosti budemo vvaΩaty, wo rank ( a ) = m ,
rank ( b ) = r , pryçomu k < m < r . Poznaçymo çerez Σ konhruencig Risa
I Im m× ∪ ∆ . Nexaj c S∈ takyj, wo rank ( c ) = k . Oskil\ky 〈 〉 ∈c a, Σ i
〈 〉 ∈a b, Θ, to 〈 〉 ∈c b, Σ Θ� . Za umovog Θ Σ� = Σ Θ� , tomu 〈 〉 ∈c b, Θ Σ� . Ce
oznaça[, wo isnu[ element d takyj, wo 〈 〉 ∈c d, Θ i 〈 〉 ∈d b, Σ . Oskil\ky
〈 〉 ∈c d, Θ i c Ik∈ , to d Ik∈ . Dali, oskil\ky 〈 〉 ∈d b, Σ , to d = b abo
〈 〉 ∈ ×d b I Im m, . Qkwo d = b , to rank ( b ) ≤ k . Supereçnist\. Qkwo Ω
〈 〉 ∈ ×d b I Im m, , to rank ( b ) ≤ m . Supereçnist\.
Takym çynom, rank ( a ) = rank ( b ) .
Teper pokaΩemo, wo 〈 〉 ∈a b H, , tobto aa−1 = bb−1
i a a−1 = b b−1 . Oskil\-
ky 〈 〉 ∈a b, Θ, to 〈 〉 ∈−a aa b, 1 Θ . Dlq bud\-qko] inversno] napivhrupy vykonu[t\-
sq nerivnist\ aa b−1 ≤ b. Qkwo prypustyty, wo aa b−1 < b, to za lemog48
rank ( )aa b−1 < rank ( b ) . Z inßoho boku, 〈 〉 ∈−b aa b, 1 Θ , tomu rank ( )aa b−1 =
= rank ( b ). Supereçnist\. OtΩe, aa b−1 = b . Zvidsy aa bb− −1 1 = bb−1. Takym
çynom, bb−1 ≤ aa−1. Qkwo prypustyty, wo bb−1 < aa−1, to rank ( b ) <
< rank ( a ) . Supereçnist\. OtΩe, aa−1 = bb−1.
Analohiçno moΩna dovesty, wo a a−1 = b b−1 . Takym çynom, 〈 〉 ∈a b H, .
1. Dereç V. D. Pro perestavni konhruenci] na antyhrupax skinçennoho ranhu // Ukr. mat. Ωurn.
– 2004. – 56, # 3. – S. 346 – 351.
2. Klyfford A., Preston H. Alhebrayçeskaq teoryq poluhrupp: V 2 t. – M.: Myr, 1972. –
T.1. – 286 s.
3. Petrich M. Inverse semigroups. – New York etc.: John Willey and Sons, 1984. – 674 p.
4. Hamilton H. Permutability of congruences on commutative semigroups // Semigroup Forum. –
1975. – 10, # 1. – P. 55 – 66.
OderΩano 16.03.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
|
| id | umjimathkievua-article-3613 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:45:47Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/99/c8837429464a2c17284dcabec40c8499.pdf |
| spelling | umjimathkievua-article-36132020-03-18T20:00:05Z Congruences of a Permutable Inverse Semigroup of Finite Rank Конгруенції переставної інверсної напівгрупи скінченного рангу Derech, V. D. Дереч, В. Д. We describe the structure of any congruence of a permutable inverse semigroup of finite rank. Описано будову будь-якої конгруенції переставної інверсної напівгрупи скінченного рангу. Institute of Mathematics, NAS of Ukraine 2005-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3613 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 4 (2005); 469–473 Український математичний журнал; Том 57 № 4 (2005); 469–473 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3613/3957 https://umj.imath.kiev.ua/index.php/umj/article/view/3613/3958 Copyright (c) 2005 Derech V. D. |
| spellingShingle | Derech, V. D. Дереч, В. Д. Congruences of a Permutable Inverse Semigroup of Finite Rank |
| title | Congruences of a Permutable Inverse Semigroup of Finite Rank |
| title_alt | Конгруенції переставної інверсної напівгрупи скінченного рангу |
| title_full | Congruences of a Permutable Inverse Semigroup of Finite Rank |
| title_fullStr | Congruences of a Permutable Inverse Semigroup of Finite Rank |
| title_full_unstemmed | Congruences of a Permutable Inverse Semigroup of Finite Rank |
| title_short | Congruences of a Permutable Inverse Semigroup of Finite Rank |
| title_sort | congruences of a permutable inverse semigroup of finite rank |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3613 |
| work_keys_str_mv | AT derechvd congruencesofapermutableinversesemigroupoffiniterank AT derečvd congruencesofapermutableinversesemigroupoffiniterank AT derechvd kongruencííperestavnoíínversnoínapívgrupiskínčennogorangu AT derečvd kongruencííperestavnoíínversnoínapívgrupiskínčennogorangu |