Structure of a permutable Munn semigroup of finite rank
A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank.
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| Date: | 2006 |
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| Main Authors: | , |
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| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3492 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509593319243776 |
|---|---|
| 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 | 2020-03-18T19:56:00Z |
| description | A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank. |
| first_indexed | 2026-03-24T02:43:34Z |
| format | Article |
| fulltext |
UDK 512.534.5
V.�D.�Dereç (Vinnyc. nac. texn. un-t)
STRUKTURA PERESTAVNO} NAPIVHRUPY MANNA
SKINÇENNOHO RANHU
A semigroup any two congruences of which commute as binary relations is called a permutable
semigroup. We describe the structure of a permutable Munn semigroup of a finite rank.
Napivhrupa, bud\-qki dvi konhruenci] qko] perestavni qk binarni vidnoßennq, nazyva[t\sq pe-
restavnog. V statti opysu[t\sq budova perestavno] napivhrupy Manna skinçennoho ranhu.
Vstup. Sered zadaç, qki rozhlqdagt\sq v teori] napivhrup, vaΩlyve misce zaj-
ma[ problematyka poßuku vza[mozv’qzkiv miΩ vlastyvostqmy reßitky konhru-
encij i vlastyvostqmy samo] napivhrupy. Zokrema, cikavog [ zadaça vstanovlen-
nq struktury ti[] çy inßo] napivhrupy, bud\-qki dvi konhruenci] qko] komutugt\
vidnosno zvyçajno] operaci] kompozyci]. Do takyx napivhrup naleΩat\ hrupy,
napivhrupy Brandta ta in.
U cij statti same taka zadaça rozhlqda[t\sq dlq napivhrupy Manna (dyv. [1]),
tobto napivhrupy vsix izomorfizmiv miΩ holovnymy idealamy napivreßitky vid-
nosno operaci] kompozyci] peretvoren\. Napivhrupy Manna vidihragt\ vaΩlyvu
rol\ u teori] zobraΩen\ inversnyx napivhrup [2, s. 170], tomu ]x klasyfikaciq za
ti[g çy inßog oznakog [ cilkom aktual\nog zadaçeg.
Osnovnym rezul\tatom dano] statti [ teorema 1, v qkij u terminax bazysno]
napivreßitky z’qsovu[t\sq budova vidpovidno] perestavno] (dyv. p.61) napivhrupy
Manna skinçennoho ranhu. V p.63 takoΩ pokazano, wo bud\-qkyj nenul\ovyj ide-
al perestavno] napivhrupy Manna skinçennoho ranhu [ wil\nym.
1. Terminolohiq i poznaçennq. Nexaj S — dovil\na napivreßitka. Pozna-
çymo çerez Φ ( S ) napivhrupu Manna, tobto napivhrupu vsix izomorfizmiv miΩ ho-
lovnymy idealamy napivreßitky S vidnosno zvyçajno] operaci] superpozyci]
binarnyx vidnoßen\.
Napivreßitku S nazyvagt\ napivreßitkog skinçenno] dovΩyny, qkwo isnu[
natural\ne çyslo n take, wo dovΩyna bud\-qkoho lancgΩka z S ne6perevywu[
n . Oçevydno, wo napivreßitka skinçenno] dovΩyny ma[ najmenßyj element —
nul\.
Vysotu elementa a napivreßitky S poznaçymo çerez rank ( a ) . Nexaj f ∈
∈ Φ( S ) . Qkwo d ( f ) = a S (tut d ( f ) — oblast\ vyznaçennq peretvorennq f ), to,
za oznaçennqm, rank ( f ) = rank ( a ) . U roboti [3] (lema 4, p. 1) dovedeno, wo funk-
ciq rank : Φ ( S ) → N0 [ ranhovog, tobto dlq bud\-qkyx f , ϕ ∈ Φ ( S ) vykonu[t\sq
nerivnist\ rank ( f � ϕ ) ≤ min ( rank ( f ) , rank ( ϕ ) ) .
Vsi inßi neobxidni ponqttq z teori] napivhrup i teori] inversnyx napivhrup
moΩna znajty vidpovidno v monohrafiqx [2] i [4].
2. Osnovnyj rezul\tat. Zrozumilo, wo budova napivhrupy Φ ( S ) cilkom
vyznaça[t\sq budovog napivreßitky S , qka, oçevydno, izomorfna napivhrupi
vsix idempotentiv napivhrupy Φ ( S ) . Tomu osnovnyj rezul\tat statti bude sfor-
mul\ovano v terminax napivreßitky S . Nexaj S — napivreßitka skinçenno] dov-
Ωyny. Budemo hovoryty, wo S zadovol\nq[ umovu D , koly vykonu[t\sq taka
vymoha: qkwo a < b (pryçomu rank ( a ) ≥ 1), to isnu[ element c takyj, wo c ≠
≠ a , c < b , rank ( c ) = rank ( a ) . Teper sformulg[mo umovu R : dlq bud\-qkoho
e ∈ S ( rank ( e ) ≥ 2 ) isnugt\ elementy b , c ∈ S taki, wo b ≠ c , b < e , c < e ,
rank ( b ) = rank ( c ) = rank ( e ) – 1.
Lema 1. Dlq napivreßitky S skinçenno] dovΩyny umovy D i R [ ekviva-
lentnymy.
Dovedennq. Prypustymo, wo vykonu[t\sq umova D . Nexaj e ∈ S , pryçomu
© V.6D.6DEREÇ, 2006
742 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
STRUKTURA PERESTAVNO} NAPIVHRUPY MANNA SKINÇENNOHO RANHU 743
rank ( e ) ≥ 2. Zrozumilo, wo isnu[ element b takyj, wo b < e i rank ( b ) =
= rank ( e ) – 1. Za6umovog D isnu[ element c takyj, wo c ≠ b , c < e , rank ( c ) =
= rank ( b ) . OtΩe, vykonu[t\sq umova R .
Navpaky, nexaj vykonu[t\sq umova R . Vyberemo a < b (tut rank ( a ) ≥ 1 ).
Vypadok A. Element a naleΩyt\ maksymal\nomu (za kil\kistg elementiv)
lancgΩku, wo z’[dnu[ 0 i b . V c\omu lancgΩku vyberemo element s takyj, wo
a ≺ s (de ≺ — znaçok pokryttq), todi rank ( s ) = rank ( a ) + 1. Za umovog R isnu-
gt\ elementy x i y taki, wo x ≠ y , x < s , y < s , rank ( x ) = rank ( y ) = rank ( a ) =
= rank ( s ) – 1. Zrozumilo, wo x ≠ a abo y ≠ a . Nexaj, napryklad, x ≠ a , krim
toho, x < b i rank ( x ) = rank ( a ) = rank ( s ) – 1. OtΩe, vykonu[t\sq umova D .
Vypadok B. Element a ne6naleΩyt\ Ωodnomu maksymal\nomu (za kil\kistg
elementiv) lancgΩku, wo z’[dnu[ 0 i b . Nexaj L — takyj maksymal\nyj lan-
cgΩok. Isnu[ element u ∈ L takyj, wo rank ( u ) = rank ( a ) + 1. Za6umovog R
znajdut\sq elementy z i ν taki, wo z ≠ ν , z < u , ν < u i rank ( z ) = rank ( ν ) =
= rank ( u ) – 1 = rank ( a ) . Zrozumilo, wo z ≠ a abo ν ≠ a . Nexaj, napryklad, z ≠
≠ a . Krim c\oho z < b i rank ( z ) = rank ( a ) . OtΩe, vykonu[t\sq umova D .
Teper sformulg[mo i dovedemo osnovnu teoremu statti.
Teorema 1. Nexaj S — napivreßitka skinçenno] dovΩyny.
Napivhrupa Manna Φ ( S ) perestavna todi i til\ky todi, koly vykonugt\sq
taki dvi umovy:
1) qkwo a ∈ S i b ∈ S , pryçomu rank ( a ) = rank ( b ) , to aS � bS ;
2) dlq bud\-qkoho e ∈ S ( rank ( e ) ≥ 2 ) isnugt\ f ∈ S i ω ∈ S taki, wo f ≠
≠ ω , f < e , ω < e i rank ( f ) = rank ( ω ) = rank ( e ) – 1.
Dovedennq. Spoçatku rozhlqnemo vypadok, koly dlq bud\-qkoho a ∈ S
rank ( a ) ≤ 1. V c\omu vypadku napivhrupa Φ ( S ) abo odnoelementna, abo [ napiv-
hrupog Brandta. Vidomo, wo bud\-qki dvi konhruenci] napivhrupy Brandta [
perestavnymy.
Dali budemo rozhlqdaty vypadok, koly v napivreßitci S isnu[ element, ranh
qkoho ne menßyj za62. Spoçatku dovedemo dostatnist\, tobto nexaj dlq napiv-
reßitky S skinçenno] dovΩyny vykonugt\sq umovy (1) i (2). Umova (1) zabez-
peçu[ linijnu vporqdkovanist\ (vidnosno vklgçennq) idealiv napivhrupy Φ ( S )
(dyv. [3], teorema61). PokaΩemo teper, wo bud\-qka konhruenciq Θ napivhrupy
Φ ( S ) ma[ formu Θ = I × I ∪ Ω , de I — ideal napivhrupy Φ ( S ) , a Ω ⊆ H ( H
— vidnoßennq Hrina).
Nexaj Θ — konhruenciq na napivhrupi Φ ( S ) . Lehko pereviryty, wo IΘ = { f ∈
∈ Φ( S ) | 〈 f ; 0 〉 ∈ Θ } [ idealom napivhrupy Φ ( S ) . Oskil\ky koΩnyj ideal napiv-
hrupy Φ ( S ) [ ranhovym (dyv. [3], teorema 1), to isnu[ natural\ne çyslo k take,
wo IΘ = Ik = { f ∈ Φ( S ) | rank ( f ) ≤ k } . Nexaj 〈 f ; ϕ 〉 ∈ Θ i rank ( f ) > k , todi, oçe-
vydno, i rank ( ϕ ) > k . PokaΩemo spoçatku, wo rank ( f ) = rank ( ϕ ) . Prypustymo
protyleΩne, tobto rank ( f ) ≠ rank (ϕ). Nexaj dlq konkretnosti rank ( f ) < rank (ϕ).
Oskil\ky 〈 f , ϕ 〉 ∈ Θ, to f f� �− −1 1, ϕ ϕ ∈ Θ. Zvidsy f f� � � �− − −1 1 1ϕ ϕ ϕ ϕ, ∈ Θ .
Rozhlqnemo moΩlyvi vypadky.
Perßyj vypadok: rank( )f f� � �− −1 1ϕ ϕ ≤ k .
Todi ( 0, ϕ � ϕ
–1
) ∈ Θ . Zvidsy ϕ � ϕ
–1 ∈ Ik , tobto rank ( ϕ � ϕ
–1
) ≤ k . Ale
rank ( ϕ � ϕ
–1
) = rank ( ϕ ) > k . Supereçnist\.
Druhyj vypadok: rank ( f � f
–1 � ϕ � ϕ
–1
) > k .
Zrozumilo, wo ma[ misce vklgçennq f � f
–1 � ϕ � ϕ
–1 ⊆ ϕ � ϕ
–1, a oskil\ky
rank ( f � f
–1 � ϕ � ϕ
–1
) ≤ rank ( f � f
–1
) = rank ( f ) < rank (ϕ) = rank ( ϕ � ϕ
–1
) , to
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
744 V.6D.6DEREÇ
f � f
–1 � ϕ � ϕ
–1 ⊂ ϕ � ϕ
–1 (strohe vklgçennq). Za umovog62 (tobto umovog R ,
qka za lemog61 ekvivalentna umovi D ) isnu[ idempotent ω ∈ Φ ( S ) takyj, wo
rank ( f � f
–1 � ϕ � ϕ
–1
) = rank (ω) , ω ⊂ ϕ � ϕ
–1 i ω ≠ f � f
–1 � ϕ � ϕ
–1. Oskil\ky
〈 f � f
–1 � ϕ � ϕ
–1, ϕ � ϕ
–1
〉 ∈ Θ , to 〈 f � f
–1 � ϕ � ϕ
–1 � ω , ϕ � ϕ
–1 � ω 〉 ∈ Θ abo
〈 f � f
–1 � ω , ω 〉 ∈ Θ . Qkwo rank ( f � f
–1 � ω ) ≤ k , to 〈 0, ω 〉 ∈ Θ. Zvidsy rank (ω) ≤
≤ k . Supereçnist\.
Qkwo Ω rank ( f � f
–1 � ω ) > k , to zastosovu[mo do vporqdkovano] pary
〈 f � f
–1 � ω , ω 〉 taki sami mirkuvannq, wo i vywe. ProdovΩugçy cej proces (a
vin, oçevydno, [ skinçennym), dlq deqkoho elementa ψ ∈ Φ( S ) ma[mo 〈 0 , ψ 〉 ∈
∈ Θ , pryçomu rank (ψ) > k . Supereçnist\. Takym çynom, rank ( f ) = rank ( ϕ ) .
Teper pokaΩemo, wo f � f
–1 = ϕ � ϕ
–1 i f
–1 � f = ϕ–1 � ϕ , tobto 〈 f , ϕ 〉 ∈ H .
Oskil\ky 〈 f , ϕ 〉 ∈ Θ , to 〈 f , f � f
–1 � ϕ 〉 ∈ Θ . Oçevydno, ma[ misce vklgçennq
f � f
–1 � ϕ ⊆ ϕ . Qkwo prypustyty, wo f � f
–1 � ϕ ⊂ ϕ (strohe vklgçennq), to
rank ( f � f
–1 � ϕ ) < rank (ϕ). Z6inßoho boku, 〈 ϕ , f � f
–1 � ϕ 〉 ∈ Θ , tomu rank ( ϕ ) =
= rank ( f � f
–1 � ϕ ) . Supereçnist\. Takym çynom, f � f
–1 � ϕ = ϕ . Zvidsy f � f
–1 �
� ϕ � ϕ
–1 = ϕ � ϕ
–1, otΩe, ϕ � ϕ
–1 ⊆ f � f
–1. Qkwo prypustyty, wo ϕ � ϕ
–1 ⊂
⊂ f � f
–1, to rank ( ϕ ) < rank ( f ) . Supereçnist\. OtΩe, f � f
–1 = ϕ � ϕ
–1. Analo-
hiçno dovodyt\sq, wo f
–1 � f = ϕ–1 � ϕ . Takym çynom, 〈 f , ϕ 〉 ∈ H .
OtΩe, idealy napivhrupy Φ ( S ) linijno vporqdkovani i bud\-qka konhruenciq
Θ ma[ formu Θ = I × I ∪ Ω (de I — ideal, a Ω ⊆ H ) . Takym çynom, za teore-
mog z p.65 [3] napivhrupa Φ ( S ) [ perestavnog.
Dovedemo neobxidnist\. Nexaj napivhrupa Φ ( S ) [ perestavnog, todi (dyv.
[5], teorema 4) ]] idealy utvorggt\ lancgΩok vidnosno vklgçennq, a otΩe (dyv.
[3], teorema 1), vykonu[t\sq umova 1 teoremy.
Teper budemo dovodyty, wo vykonu[t\sq j umova 2. Dovedennq provedemo vid
suprotyvnoho, tobto prypustymo, wo isnu[ idempotent ω ∈ Φ ( S ) , dlq qkoho
umova62 ne6vykonu[t\sq. Nexaj rank ( ω ) = k + 1, de k + 1 ≥ 2. Rozhlqnemo na
Φ ( S ) binarne vidnoßennq Θ = Ik –1 × Ik –1 ∪ ρ ∪ ρ–1 ∪ ∆ , de Ik –1 = { f ∈ Φ( S ) |
rank ( f ) ≤ k – 1 } , ∆ = { 〈 f , f 〉 | f ∈ Φ( S ) } , ρ = { 〈 f , ϕ 〉 ∈ Φ( S ) | f ⊂ ϕ ∧ rank ( f ) =
= k ∧ rank ( ϕ ) = k + 1} . Oçevydno, wo binarne vidnoßennq Θ [ refleksyvnym i
symetryçnym. Dali budemo dovodyty dvostoronng stabil\nist\ binarnoho vidno-
ßennq Θ . Dovedennq rozib’[mo na kil\ka lem.
Lema 2. Nexaj 〈 f , ϕ 〉 ∈ Θ , pryçomu f ⊂ ϕ , rank ( f ) = k i rank ( ϕ ) = k + 1.
Qkwo rank ( f � ψ ) = k , to 〈 f � ψ , ϕ � ψ 〉 ∈ Θ .
Qkwo Ω rank ( ψ � f ) = k , to 〈 ψ � f , ψ � ϕ 〉 ∈ Θ .
Dovedennq. Oskil\ky za umovog f ⊂ ϕ , to f � ψ ⊆ ϕ � ψ . Qkwo f � ψ =
= ϕ � ψ , to 〈 f � ψ , ϕ � ψ 〉 ∈ Θ . Qkwo Ω f � ψ ⊂ ϕ � ψ , to k = rank ( f � ψ ) <
< rank ( ϕ � ψ ) ≤ rank ( ϕ ) = k + 1 . OtΩe, rank ( ϕ � ψ ) = k + 1 . Takym çynom, 〈 f � ψ ,
ϕ � ψ 〉 ∈ ρ ⊂ Θ .
Druha çastyna lemy dovodyt\sq analohiçno.
Lema 3. Qkwo rank ( f ) = rank ( f � ϕ ) = k , to f
–1 � f ⊆ ϕ � ϕ–1
.
Dovedennq. MoΩlyvi try vypadky: a) ϕ � ϕ–1 ⊂ f
–1 � f ; b) ϕ � ϕ–1 � f
–1 � f ;
v) f
–1 � f ⊆ ϕ � ϕ–1.
Prypustymo, wo ϕ � ϕ–1 ⊂ f
–1 � f , todi k = rank ( f � ϕ ) = rank ( f � ϕ � ϕ–1
) ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
STRUKTURA PERESTAVNO} NAPIVHRUPY MANNA SKINÇENNOHO RANHU 745
≤ rank ( ϕ � ϕ–1
) < rank ( f � f
–1
) = k . Supereçnist\.
Nexaj teper ϕ � ϕ–1 � f
–1 � f , todi f
–1 � f � ϕ � ϕ–1 ⊂ f
–1 � f . PokaΩemo, wo
rank ( f
–1 � f � ϕ � ϕ–1
) = k . Spravdi,
k = rank ( f � ϕ ) = rank ( f � f
–1 � f � ϕ � ϕ–1 � ϕ ) ≤ rank ( f
–1 � f � ϕ � ϕ–1
) .
OtΩe, k ≤ rank ( f
–1 � f � ϕ � ϕ–1
) ≤ k , tobto rank ( f
–1 � f � ϕ � ϕ–1
) = k . Ale
f
–1 � f � ϕ � ϕ–1 ⊂ f
–1 � f , tomu k = rank ( f
–1 � f � ϕ � ϕ–1
) < rank ( f
–1 � f ) = k . Supe-
reçnist\.
Zalyßa[t\sq odna moΩlyvist\ f
–1 � f ⊆ ϕ � ϕ–1.
Lema 4. Qkwo rank ( f ) = rank ( ϕ � f ) , to f � f
–1 ⊆ ϕ–1 � ϕ .
Dovedennq analohiçne dovedenng poperedn\o] lemy.
Lema 5. Nexaj f ⊂ ϕ , krim toho, rank ( f ) = k i rank ( ϕ ) = k + 1. Qkwo
rank ( ϕ � ψ ) = k + 1, to rank ( f � ψ ) = k .
Dovedennq. Oskil\ky rank ( ϕ � ψ ) = rank ( ϕ ) , to za lemog63 ϕ–1
� ϕ ⊆ ψ � ψ–1.
Zvidsy f
–1
� f � ϕ–1
� ϕ ⊆ f
–1
� f � ψ � ψ–1. Ale f
–1
� f � ϕ–1
� ϕ ⊆ f
–1 � f , tomu f
–1 � f ⊆
⊆ f
–1 � f � ψ � ψ–1. OtΩe, k = rank ( f
–1 � f ) ≤ rank ( f
–1 � f � ψ � ψ–1
) ≤ rank ( f � ψ ) .
Z6inßoho boku, rank ( f � ψ ) ≤ rank ( f ) = k . Takym çynom, rank ( f � ψ ) = k .
Lema 6. Nexaj f ⊂ ϕ , pryçomu rank ( f ) = k i rank ( ϕ ) = k + 1. Qkwo
rank ( ψ � ϕ ) = k + 1, to rank ( ψ � f ) = k .
Dovedennq analohiçne dovedenng poperedn\o] lemy.
Perejdemo do dovedennq pravostoronn\o] stabil\nosti binarnoho vidnoßennq Θ .
Nexaj 〈 f , ϕ 〉 ∈ Θ . Qkwo 〈 f , ϕ 〉 ∈ Ik –1 × Ik –1 abo f = ϕ , to dovodyty nema[
çoho.
Rozhlqnemo vypadok, koly 〈 f , ϕ 〉 ∈ ρ , tobto f ⊂ ϕ , rank ( f ) = k , rank ( ϕ ) =
= k + 1.
Nexaj ψ ∈ Φ ( S ) .
A. Qkwo rank ( f � ψ ) = k , to za lemog62 〈 f � ψ , ϕ � ψ 〉 ∈ Θ .
B. Nexaj teper rank ( f � ψ ) < k , todi za lemog65 rank ( ϕ � ψ ) ≤ k . Prypusty-
mo, wo rank ( ϕ � ψ ) = k . Rozhlqnemo idempotenty f � f
–1, ϕ � ϕ–1, ϕ � ψ � ψ–1 � ϕ–1.
Zrozumilo, wo magt\ misce strohi vklgçennq f � f
–1 ⊂ ϕ � ϕ–1 i ϕ � ψ �
� ψ–1 � ϕ–1 ⊂ ϕ � ϕ–1, krim toho, rank ( f � f
–1
) = rank ( ϕ � ψ � ψ–1 � ϕ–1
) = k i
rank ( ϕ � ϕ–1
) = k + 1.
Qkwo prypustyty, wo ϕ � ψ � ψ–1 � ϕ–1 ≠ f � f
–1, to, vykorystovugçy umovu61
(dyv. formulgvannq teoremy) i naße osnovne prypuwennq (pro te, wo dlq idem-
potenta ω ∈ Φ ( S ) umova62 teoremy ne vykonu[t\sq), oderΩu[mo supereçnist\.
Teper prypustymo, wo ϕ � ψ � ψ–1 � ϕ–1 = f � f
–1
. Oskil\ky f ⊂ ϕ , to f =
= f � ϕ–1
� ϕ i f � ϕ–1 = f � f
–1
. Zvidsy f � ϕ–1
� ϕ = f � f
–1 � ϕ = f . DomnoΩyvßy riv-
nist\ ϕ � ψ � ψ–1 � ϕ–1 = f � f
–1 zliva na f � f
–1
, oderΩymo f � f
–1 � ϕ � ψ �
� ψ–1 � ϕ–1 = f � f
–1 abo f � ψ � ψ–1 � ϕ–1 = f � f
–1
.
OtΩe, k = rank ( f � f
–1
) = rank ( f � ψ � ψ–1 � ϕ–1
) ≤ rank ( f � ψ ) < k . Supereç-
nist\. Takym çynom, rank ( ϕ � ψ ) < k . Zvidsy 〈 f � ψ , ϕ � ψ 〉 ∈ Ik –1 × Ik –1 ⊆ Θ ,
tobto binarne vidnoßennq Θ [ stabil\nym sprava.
Analohiçno dovodyt\sq stabil\nist\ zliva.
OtΩe, binarne vidnoßennq Θ [ refleksyvnym, symetryçnym i stabil\nym.
Dali, poznaçymo çerez Θt tranzytyvne zamykannq binarnoho vidnoßennq Θ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
746 V.6D.6DEREÇ
Lehko perekonatysq, wo Θt — konhruenciq. Z [3] (teorema 3 z p.65) vidomo, wo
napivhrupa Φ ( S ) perestavna todi i til\ky todi, koly ]] idealy linijno vporqdko-
vani i koΩna konhruenciq ma[ formu Θ = I × I ∪ Ω , de I — ideal napivhrupy
Φ ( S ) , a Ω ⊆ H ( H — vidnoßennq Hrina). Konhruenciq Θt, qku my skonstrug-
valy vyxodqçy z prypuwennq, wo umova62 (dyv. formulgvannq teoremy) ne
vykonu[t\sq, ne pidpada[ pid navedenu vywe formu konhruenci] perestavno]
napivhrupy Manna. Spravdi, { ρ ∈ Φ( S ) | 〈 ρ , 0 〉 ∈ Θt
} = Ik –1 , krim toho, isnu[
para 〈 f , ϕ 〉 ∈ Θt taka, wo rank ( f ) = k i rank ( ϕ ) = k + 1, tobto 〈 f , ϕ 〉 ∉ H .
OderΩana supereçnist\ i zaverßu[ dovedennq teoremy.
3. Pro wil\ne ideal\ne rozßyrennq. Ideal\ne rozßyrennq S napivhrupy
V nazyva[t\sq wil\nym, qkwo bud\-qka konhruenciq na S , obmeΩennq qko] na
V [ rivnistg, takoΩ [ vidnoßennqm rivnosti.
Teorema 2. Nexaj S — napivreßitka skinçenno] dovΩyny taka, wo napiv-
hrupa Manna Φ( S ) [ perestavnog.
Qkwo konhruenciq Θ napivhrupy Φ( S ) totoΩna na ideali Ik –1 = { f ∈ Φ( S ) |
rank ( f ) ≤ k – 1 } ( de k ≥ 2 ) , to vona totoΩna i na ideali Ik = { f ∈ Φ( S ) |
rank ( f ) ≤ k } .
Dovedennq. Oskil\ky za umovog napivhrupa Φ( S ) [ perestavnog, to z teo-
remy p.65 (dyv. [3, s. 350]) bezposeredn\o vyplyva[, wo Θ ⊆ H (de H — vidno-
ßennq Hrina). PokaΩemo, wo konhruenciq Θ [ totoΩnog na ideali Ik .
Nexaj 〈 f , ϕ 〉 ∈ Θ , pryçomu rank ( f ) = k . Oskil\ky 〈 f , ϕ 〉 ∈ H , to rank ( f ) =
= rank ( ϕ ) = k . Bil\ß toho, d ( f ) = d ( ϕ ) i r ( f ) = r ( ϕ ) (tut d ( f ) i r ( f ) — vidpo-
vidni oblast\ vyznaçennq i mnoΩyna znaçen\ peretvorennq f ) . Nexaj d ( f ) =
= d ( ϕ ) = a S i r ( f ) = r ( ϕ ) = b S . Oskil\ky f i ϕ — izomorfizmy miΩ holovnymy
idealamy a S i b S , to a f = a ϕ = b .
Nexaj teper c < a , todi rank ( c ) < rank ( a ) . Zvidsy rank ( ∆cS ) ≤ k – 1 (çerez
∆cS my poznaça[mo totoΩnyj avtomorfizm holovnoho idealu c S ) . Dali, 〈 ∆cS �
� f , ∆cS � ϕ 〉 ∈ Θ, tomu wo 〈 f , ϕ 〉 ∈ Θ . Oskil\ky rank ( ∆cS � f ) ≤ k – 1 i
rank ( ∆cS � ϕ ) ≤ k – 1, to ∆cS � f = ∆cS � ϕ . Zvidsy c f = c ϕ , tobto f = ϕ .
Naslidok. Qkwo S — napivreßitka skinçenno] dovΩyny taka, wo napiv-
hrupa Φ( S ) [ perestavnog, to Φ( S ) [ wil\nym ideal\nym rozßyrennqm bud\-
qkoho svoho idealu, vidminnoho vid nulq.
1. Munn W. D. Fundamental inverse semigroups // Quart. J. Math. – 1970. – 21. – P. 157 – 170.
2. Petrich M. Inverse semigroups. – New York etc.: John Willey and Sons, 1984. – 674 p.
3. Dereç1V. D. Pro perestavni konhruenci] na antyhrupax skinçennoho ranhu // Ukr. mat. Ωurn.
– 2004. – 56, #63. – S.6346 – 351.
4. Klyfford1A., Preston1H. Alhebrayçeskaq teoryq poluhrupp. – M.: Myr, 1972. – T.61. –
2866s.
5. Hamilton H. Permutability of congruences on commutative semigroups // Semigroup Forum. –
1975. – 10, # 1. – P. 55 – 66.
OderΩano 10.12.2004,
pislq doopracgvannq — 15.02.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
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| id | umjimathkievua-article-3492 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:43:34Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0c/e1a3bc96d1eedc043e032a52db02c70c.pdf |
| spelling | umjimathkievua-article-34922020-03-18T19:56:00Z Structure of a permutable Munn semigroup of finite rank Структура переставної напівгрупи Манна скінченного рангу Derech, V. D. Дереч, В. Д. A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank. Напівгрупа, будь-які дві конгруенції якої переставні як бінарні відношення, називається переставною. В статті описується будова переставної напівгрупи Манна скінченного рангу. Institute of Mathematics, NAS of Ukraine 2006-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3492 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 6 (2006); 742–746 Український математичний журнал; Том 58 № 6 (2006); 742–746 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3492/3721 https://umj.imath.kiev.ua/index.php/umj/article/view/3492/3722 Copyright (c) 2006 Derech V. D. |
| spellingShingle | Derech, V. D. Дереч, В. Д. Structure of a permutable Munn semigroup of finite rank |
| title | Structure of a permutable Munn semigroup of finite rank |
| title_alt | Структура переставної напівгрупи Манна скінченного рангу |
| title_full | Structure of a permutable Munn semigroup of finite rank |
| title_fullStr | Structure of a permutable Munn semigroup of finite rank |
| title_full_unstemmed | Structure of a permutable Munn semigroup of finite rank |
| title_short | Structure of a permutable Munn semigroup of finite rank |
| title_sort | structure of a permutable munn semigroup of finite rank |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3492 |
| work_keys_str_mv | AT derechvd structureofapermutablemunnsemigroupoffiniterank AT derečvd structureofapermutablemunnsemigroupoffiniterank AT derechvd strukturaperestavnoínapívgrupimannaskínčennogorangu AT derečvd strukturaperestavnoínapívgrupimannaskínčennogorangu |