Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental
We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental.
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| Дата: | 2009 |
|---|---|
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| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509013301526528 |
|---|---|
| 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 | 2020-03-18T19:43:07Z |
| description | We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental. |
| first_indexed | 2026-03-24T02:34:21Z |
| format | Article |
| fulltext |
UDK 512.534.5
V. D. Dereç (Vinnyc. nac. texn. un-t)
STRUKTURA NAPIVHRUPY MANNA SKINÇENNOHO RANHU,
KOÛNYJ STABIL|NYJ PORQDOK QKO} {
FUNDAMENTAL|NYM ABO ANTYFUNDAMENTAL|NYM
We describe the structure of a Munn semigroup of finite rank whose every stable order is fundamental or
antifundamental.
Opys¥vaetsq struktura poluhrupp¥ Manna koneçnoho ranha, kaΩd¥j stabyl\n¥j porqdok ko-
toroj fundamental\n¥j yly antyfundamental\n¥j.
Napivhrupa Manna (dyv. [1]), tobto inversna napivhrupa vsix izomorfizmiv miΩ
holovnymy idealamy napivreßitky vidnosno zvyçajno] operaci] kompozyci] binar-
nyx vidnoßen\, vidihra[ fundamental\nu rol\ u teori] zobraΩen\ inversnyx na-
pivhrup (dyv. [2, s. 170]). Tomu riznobiçne vyvçennq takyx napivhrup i ]x klasy-
fikaciq [ cilkom aktual\nog zadaçeg.
V danij roboti my z�qsovu[mo strukturu napivhrupy Manna skinçennoho ran-
hu, koΩnyj stabil\nyj porqdok qko] [ fundamental\nym abo antyfundamen-
tal\nym (oznaçennq dyv. v p. 1). Osnovnym rezul\tatom statti [ teorema 1.
1. Osnovni oznaçennq i terminolohiq. Napivreßitka E nazyva[t\sq napiv-
reßitkog skinçenno] dovΩyny, qkwo isnu[ natural\ne çyslo n take, wo dov-
Ωyna bud\-qkoho lancgΩka z E ne perevywu[ çysla n.
Nexaj E � napivreßitka skinçenno] dovΩyny. Oçevydno vona mistyt\ naj-
menßyj element, qkyj poznaça[mo çerez 0. Çerez TE poznaçymo napivhrupu
Manna, tobto inversnu napivhrupu vsix izomorfizmiv miΩ holovnymy idealamy
napivreßitky E vidnosno zvyçajno] operaci] kompozyci] binarnyx vidnoßen\.
Zrozumilo, wo peretvorennq
0
0
[ najmenßym elementom napivhrupy TE . Ob-
last\ vyznaçennq i mnoΩynu znaçen\ peretvorennq f TE∈ budemo vidpovidno
poznaçaty çerez dom( )f i im( )f .
Nexaj S � dovil\na napivhrupa, a N0 � mnoΩyna vsix nevid�[mnyx cilyx
çysel. Funkcig rank : S → N0 nazyvagt\ ranhovog na napivhrupi S, qkwo dlq
bud\-qkyx a, b S∈ vykonu[t\sq nerivnist\ rank( )ab ≤ min ( )rank a( , rank( )b ).
Çyslo rank( )x nazyvagt\ ranhom elementa x.
Nexaj S � inversna napivhrupa, napivreßitka idempotentiv qko] ma[ skinçen-
nu dovΩynu. Funkciq rank( )a = h aa–1( ) , de h aa–1( ) � vysota idempotenta
aa–1 u napivreßitci idempotentiv napivhrupy S, [ ranhovog funkci[g (dyv.
[3]). Budemo hovoryty, wo inversna napivhrupa [ napivhrupog skinçennoho ranhu,
qkwo napivreßitka ]] idempotentiv ma[ skinçennu dovΩynu.
Nexaj f TE∈ . Qkwo dom( )f = aE, to, za oznaçennqm, rank( )f = rank( )a
(de rank( )a � vysota elementa a v napivreßitci E). V statti [4] dovedeno, wo
funkciq rank : TE → N0 [ ranhovog.
Vidnoßennq porqdku τ na dovil\nij napivhrupi S nazyva[t\sq fundamen-
tal\nym (dyv. [5] abo [6, s. 289]), qkwo vporqdkovana napivhrupa ( ; )S τ O -izo-
morfna deqkij napivhrupi çastkovyx peretvoren\ mnoΩyny, qka vporqdkovana
vidnoßennqm vklgçennq. Qkwo τ � fundamental\ne vidnoßennq porqdku na
napivhrupi S, to vidnoßennq porqdku τ–1 nazyvagt\ antyfundamental\nym.
Ohlqd rezul\tativ pro fundamental\ni porqdky na inversnyx napivhrupax moΩ-
na znajty v [6].
Nexaj P � vporqdkovana mnoΩyna z najmenßym elementom 0. Çerez p bu-
© V. D. DEREÇ, 2009
52 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
STRUKTURA NAPIVHRUPY MANNA SKINÇENNOHO RANHU … 53
demo poznaçaty vidnoßennq pokryttq. Qkwo 0 p a , to element a nazyvagt\
atomom vporqdkovano] mnoΩyny P. Qkwo E � netryvial\na napivreßitka skin-
çenno] dovΩyny, to, oçevydno, vona mistyt\ atomy. KaΩut\, wo element b E∈
[ ob�[dnannqm atomiv, qkwo isnu[ pidmnoΩyna C mnoΩyny atomiv taka, wo
supC = b.
Ideal I napivhrupy S nazyvagt\ wil\nym, qkwo bud\-qkyj homomorfizm na-
pivhrupy S, in�[ktyvnyj na ideali I, [ in�[ktyvnym na vsij napivhrupi S.
Vsi inßi neobxidni ponqttq z teori] napivhrup moΩna znajty v [7].
2. Osnovnyj rezul\tat. Zrozumilo, wo struktura napivhrupy Manna TE
cilkom odnoznaçno vyznaça[t\sq strukturog napivreßitky E. Tomu osnovnyj
rezul\tat statti my sformulg[mo v terminax napivreßitky E.
Spoçatku rozhlqnemo najprostißyj vypadok, a same, nexaj dlq bud\-qkoho
elementa b E∈ vykonu[t\sq nerivnist\ rank( )b ≤ 1. Qkwo rank( )b = 0 dlq
bud\-qkoho elementa b E∈ , to napivreßitka E tryvial\na, a otΩe, napivhrupa
Manna TE teΩ tryvial\na. Qkwo Ω u napivreßitci E isnu[ element, ranh qko-
ho 1, to napivhrupa TE , oçevydno, [ napivhrupog Brandta, pryçomu ]] struktur-
na hrupa odnoelementna. Stabil\ni porqdky napivhrupy Brandta z tryvial\nog
strukturnog hrupog duΩe lehko opysugt\sq (dyv. nastupnu lemu). Cej re-
zul\tat moΩna vvaΩaty matematyçnym fol\klorom.
Lema 1. Qkwo B [ napivhrupog Brandta, strukturna hrupa qko] tryvial\-
na, to stabil\ni porqdky napivhrupy B vyçerpugt\sq takymy: ∆, 0{ } × B U ∆,
B × 0{ } U ∆, de 0 � nul\ napivhrupy Brandta, a ∆ � totoΩne peretvorennq
na nij.
Dovedennq. Lehko perevirq[t\sq.
Sytuaciq znaçno uskladng[t\sq, qkwo v napivreßitci E isnu[ prynajmni
odyn element x takyj, wo rank( )x ≥ 2. Nadali budemo rozhlqdaty same taki na-
pivreßitky.
Teper sformulg[mo osnovnyj rezul\tat statti.
Teorema 1. Nexaj E � napivreßitka skinçenno] dovΩyny, TE � vidpovidna
napivhrupa Manna.
Nastupni umovy [ ekvivalentnymy:
1) bud\-qkyj stabil\nyj porqdok na napivhrupi TE [ fundamental\nym abo
antyfundamental\nym;
2) koΩnyj nenul\ovyj element napivreßitky E [ ob�[dnannqm atomiv;
3) ideal I1 = f TE∈{ rank( )f ≤ 1} [ wil\nym u napivhrupi TE .
Dlq dovedennq teoremy nam znadoblqt\sq dekil\ka lem.
Lema 2. Nexaj u napivreßitci P elementy a, b, c taki, wo a cp , b cp
i a b≠ , todi c = sup ,a b{ } .
Dovedennq. Nexaj element m P∈ takyj, wo a m≤ i b m≤ , todi am = a,
bm = b, krim toho, za umovog ac = a i bc = b. Zvidsy acm = am = a. Tobto a ≤
≤ cm. Prypustymo, wo a = cm. Oskil\ky bc = b, to bcm = bm = b abo ba = b,
tobto b a≤ .
1. Qkwo a = b, to ce supereçyt\ umovi.
2. Qkwo b a< , to b < a < c, wo supereçyt\ umovi.
OtΩe, a < cm, ale Ω cm ≤ c. Oskil\ky a cp , to cm = c. Ostannq rivnist\
oznaça[, wo c m≤ . Takym çynom, c = sup ,a b{ } .
Lemu 2 dovedeno.
Lema 3. Qkwo v napivreßitci P skinçenno] dovΩyny isnu[ nenul\ovyj ele-
ment, qkyj ne [ ob�[dnannqm atomiv, to isnu[ element c P∈ , qkyj pokryva[
toçno odyn nenul\ovyj element.
Dovedennq. Poznaçymo çerez K mnoΩynu nenul\ovyx elementiv, koΩnyj z
qkyx ne moΩna podaty qk ob�[dnannq atomiv. Vyberemo u mnoΩyni K element
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
54 V. D. DEREÇ
(poznaçymo joho çerez c), ranh qkoho najmenßyj sered usix elementiv mnoΩyny
K. Oçevydno, wo rank( )c ≥ 2. PokaΩemo, wo element c pokryva[ toçno odyn
nenul\ovyj element. Prypustymo protyleΩne, tobto element c pokryva[ dva
riznyx elementy (skaΩimo, x i y). Oskil\ky rank( )x < rank( )c i rank( )y <
< rank( )c , to koΩnyj z elementiv x i y [ ob�[dnannqm atomiv. Za poperedn\og
lemog c = sup ,x y{ } = x y∨ . Oskil\ky x = sup A1 i y = sup A2, de A A1 ⊆ i
A A2 ⊆ (tut çerez A my poznaça[mo mnoΩynu vsix atomiv napivreßitky P), to
c = x y∨ = sup A1 ∨ sup A2 = sup A A1 2U( ) . Tobto element c [ ob�[dnannqm ato-
miv. Supereçnist\.
Lemu 3 dovedeno.
Lema 4. Nexaj E � napivreßitka skinçenno] dovΩyny. Qkwo element c
pokryva[ toçno odyn nenul\ovyj element b , to A c( ) = A b( ) ( de A c( ) i
A b( ) � atomy elementiv c i b vidpovidno).
Dovedennq. Oskil\ky b cp , to A b( ) � A c( ) . Nexaj a A c∈ ( ) , todi a < c.
Rozhlqnemo bud\-qkyj maksymal\nyj lancgΩok, wo z�[dnu[ a i c . U c\omu
lancgΩku [ element m takyj, wo m ≠ 0 i m cp . Za umovog element c po-
kryva[ toçno odyn nenul\ovyj element (a same, element b), tomu m = b. Zvidsy
a ≤ b. OtΩe, A c( ) � A b( ). Takym çynom, A c( ) = A b( ).
Lemu 4 dovedeno.
Dali, nexaj E � napivreßitka skinçenno] dovΩyny, qka mistyt\ nenul\ovyj
element, wo ne [ ob�[dnannqm atomiv. Todi za lemog 3 isnu[ element c
( rank( )c ≥ 2), qkyj pokryva[ toçno odyn element b. Poznaçymo çerez ∆c i ∆b
vidpovidno totoΩni peretvorennq holovnyx idealiv cE i bE . Oçevydno, wo
∆b � ∆c . Na napivhrupi Manna TE budemo rozhlqdaty binarne vidnoßennq ρ =
= f{ ° ∆c ° ϕ, f bo ∆ ° ϕ f, ϕ ∈ }TE . Poznaçymo çerez ρt tranzytyvne zamy-
kannq binarnoho vidnoßennq ρ.
Lema 5. Qkwo ( , )α β ∈ ρt , to β α⊆ .
Dovedennq. Oskil\ky ρt [ tranzytyvnym zamykannqm binarnoho vidnoßen-
nq ρ, to isnugt\ τ1, τ2, … , τn ET∈ taki, wo ( , )α τ1 ∈ ρ, ( , )τ τ1 2 ∈ ρ, … , ( –τn 1,
τn) ∈ ρ i ( , )τ βn ∈ ρ. Oskil\ky ∆b � ∆c , to f bo ∆ o ϕ � f ° ∆c ° ϕ dlq bud\-
qkyx f, ϕ ∈TE . OtΩe, β � τn � τn –1 � … � τ2 � τ1 � α . Takym çynom,
β α⊆ .
Lemu 5 dovedeno.
Teper my moΩemo dovesty implikacig 1 → 2. OtΩe, nexaj bud\-qkyj sta-
bil\nyj porqdok na napivhrupi TE [ fundamental\nym abo antyfundamental\-
nym. Nam treba dovesty, wo bud\-qkyj nenul\ovyj element napivreßitky E [
ob�[dnannqm atomiv. Prypustymo protyleΩne, tobto v napivreßitci E isnu[ ne-
nul\ovyj element, qkyj ne moΩna podaty u vyhlqdi ob�[dnannq atomiv. Todi za
lemog 3 isnu[ element c, qkyj pokryva[ toçno odyn nenul\ovyj element (ska-
Ωimo, b). Tobto b cp , pryçomu rank( )c ≥ 2. Rozhlqnemo na TE binarne vidno-
ßennq Σ = 0
0( ){ } × I1 U ρt U ∆, de ρ = f{ ° ∆c ° ϕ, f bo ∆ ° ϕ f, ϕ ∈ }TE , I1 =
= f TE∈{ rank( )f ≤ 1} , ∆ = ψ ψ,{ ψ ∈ }TE . Oçevydno, wo binarne vidnoßennq
Σ [ refleksyvnym. Dali, binarne vidnoßennq ρ, oçevydno, [ stabil\nym. Leh-
ko pokazaty, wo tranzytyvne zamykannq stabil\noho binarnoho vidnoßennq teΩ
[ stabil\nym. OtΩe, ρt � stabil\ne binarne vidnoßennq. Zvidsy vyplyva[, wo
Σ [ takoΩ stabil\nym binarnym vidnoßennqm. Teper pokaΩemo tranzytyvnist\
binarnoho vidnoßennq Σ. Rozhlqnemo moΩlyvi vypadky.
Vypadok 1. ( , )α β ∈ ρt i ( , )β γ ∈ ρt .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
STRUKTURA NAPIVHRUPY MANNA SKINÇENNOHO RANHU … 55
Oskil\ky ρt � tranzytyvne binarne vidnoßennq, to ( , )α γ ∈ ρt .
Vypadok 2. ( , )α β ∈ ρt i ( , )β γ ∈
0
0
× I1.
Todi β =
0
0
. PokaΩemo, wo α =
0
0
. Oskil\ky ( , )α β ∈ ρt , to isnugt\
τ1, τ2, … , τn ET∈ taki, wo ( , )α τ1 ∈ ρ, ( , )τ τ1 2 ∈ ρ, … , ( –τn 1, τn) ∈ ρ, ( , )τ βn ∈
∈ ρ. Oskil\ky ( , )τ βn ∈ ρ, to isnugt\ f, ϕ ∈TE taki, wo τn = f ° ∆c ° ϕ i β =
= f ° ∆b ° ϕ. Nexaj im( )f = mE i dom( )ϕ = kE. PokaΩemo, wo mE I bE I kE =
= mbkE = 0{ }. Prypustymo protyleΩne, tobto isnu[ element z takyj, wo z ≠ 0
i z ∈ mE I bE I kE. Zvidsy vyplyva[, wo isnugt\ nenul\ovi elementy x, y E∈
taki, wo
x
z
f
∈ ,
z
z b
∈∆ i
z
y
∈ϕ . OtΩe,
x
y
∈ f ° ∆b ° ϕ, pryçomu
x
y
≠
≠
0
0
. Supereçnist\. Takym çynom, mE I bE I kE = mbkE = 0{ }, zvidky mbk =
= 0. Oskil\ky b cp , pryçomu element c pokryva[ [dynyj element, to zvidsy
lehko vyplyva[, wo mck = 0 i m E I cE I kE = 0{ }. OtΩe, f ° ∆c ° ϕ =
0
0
=
= τn. Oskil\ky ( –τn 1, τn) ∈ ρ i τn =
0
0
, to, qk i vywe, dovodymo, wo τn –1 =
=
0
0
. Analohiçno τn –1 = … = τ2 = τ1 = α =
0
0
. Takym çynom, u druhomu vy-
padku α = β =
0
0
. OtΩe, ( , )a γ ∈Σ .
Vypadok 3. ( , )α β ∈
0
0
× I1 i ( , )β γ ∈ ρt .
Todi α =
0
0
i β ∈ I1 . Qkwo β =
0
0
, to, qk lehko pokazaty, γ =
0
0
. Ne-
xaj teper β =
0
0
1
2
a
a
, de a1 i a2 � atomy napivreßitky E. Oskil\ky (β ,
γ ) ∈ ρt , to isnugt\ µ1, µ2, … , µn ET∈ taki, wo ( , )β µ1 ∈ ρ, ( , )µ µ1 2 ∈ ρ, …
… , ( –µn 1, µn) ∈ ρ, ( , )µ γn ∈ ρ. Dali, ( , )β µ1 ∈ ρ i β =
0
0
1
2
a
a
. PokaΩemo, wo
µ1 = β. Oskil\ky ( , )β µ1 ∈ ρ, to isnugt\ f, ϕ ∈TE taki, wo β = f ° ∆c ° ϕ i
µ1 = f ° ∆b ° ϕ. Oskil\ky ∆ ∆b c⊂ , to µ β1 ⊆ . Dali, β =
0
0
1
2
a
a
= f ° ∆c ° ϕ,
tomu isnu[ atom a E∈ takyj, wo
0
0
1a
a
� f,
0
0
a
a
� ∆c ,
0
0 2
a
a
� ϕ. Oçe-
vydno,
0
0
a
a
[ atomom napivreßitky E TE( ), pryçomu
0
0
a
a
� ∆c . Oskil\ky
∆b p ∆c , to za lemog 4
0
0
a
a
� ∆b . Zvidsy vyplyva[, wo
0
0
1
2
a
a
� µ1. Krim
toho, qk vΩe zaznaçalosq, µ β1 ⊆ , tomu µ1 =
0
0
1
2
a
a
= β. Oskil\ky ( , )µ µ1 2 ∈
∈ ρ, ( , )µ µ2 3 ∈ ρ, … , ( , )µ γn ∈ ρ, to, mirkugçy analohiçnym çynom, oderΩu[mo
µ2 = µ3 = … = µn = γ = β. OtΩe, ( , )α γ ∈Σ .
Lemu 5 dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
56 V. D. DEREÇ
Teper pokaΩemo, wo binarne vidnoßennq Σ [ antysymetryçnym.
Rozhlqnemo moΩlyvi vypadky.
Vypadok A. ( , )α β ∈ ρt i ( , )β α ∈ ρt .
Todi za lemog 5 β α⊆ i α β⊆ . Zvidsy α = β.
Vypadok B. ( , )α β ∈ ρt i ( , )β α ∈
0
0
× I1. Oskil\ky β =
0
0
, to, qk bu-
lo dovedeno vywe, α =
0
0
. Tobto α = β. Takym çynom, binarne vidnoßennq Σ
[ stabil\nym porqdkom na TE . Vidomo [6, s. 303], wo stabil\nyj porqdok Ω na
inversnij napivhrupi [ fundamental\nym todi i til\ky todi, koly Ω ⊆ ω (de ω
� kanoniçnyj porqdok). Ale oçevydno, wo Σ ⊄ ω i Σ ⊄ ω–1. Supereçnist\.
Dali ob©runtu[mo implikacig 2 → 1. A same, nexaj bud\-qkyj nenul\ovyj
element napivreßitky E [ ob�[dnannqm atomiv. Nam treba dovesty, wo bud\-
qkyj stabil\nyj porqdok na inversnij napivhrupi TE [ fundamental\nym abo
antyfundamental\nym. Pered tym qk perejty do dovedennq sformulg[mo
kil\ka potribnyx rezul\tativ.
Rezul\tat 1 [8, s. 564]. Dlq idealu I inversno] napivhrupy nastupni vlasty-
vosti [ ekvivalentnymy:
1) I � livoreduktyvnyj ideal;
2) I � pravoreduktyvnyj ideal;
3) I � reduktyvnyj ideal.
Rezul\tat 2 [8, s. 564]. Dlq idealu I inversno] napivhrupy S nastupni
vlastyvosti [ ekvivalentnymy:
1) I [ wil\nym idealom;
2) I [ ∨-bazysnym idealom;
3) I [ reduktyvnym idealom.
(Druha vlastyvist\ oznaça[, wo koΩnyj element b S∈ moΩna podaty u vyhlqdi
b = sup A, de A I⊆ ).
Dali, nexaj S � inversna napivhrupa skinçennoho ranhu z nulem 0, b S∈ �
dovil\nyj element napivhrupy S. Poznaçymo çerez R b1( ) mnoΩynu x S∈{ x ≤
≤ b ∧ rank (x) ≤ 1} .
Rezul\tat 3 [9]. Nexaj S � inversna napivhrupa skinçennoho ranhu z nu-
lem. Homomorfizm F : b � R b1( ) [ izomorfizmom todi i til\ky todi, koly ideal
I1 = x S∈{ rank (x) ≤ 1} [ wil\nym.
Rezul\tat 4 [9]. Nexaj S � inversna napivhrupa skinçennoho ranhu z nu-
lem. Qkwo ideal I1 = f TE∈{ rank (f) ≤ 1} [ wil\nym, to ma[ misce ekvivalent-
nist\ R b1( ) � R c1( ) � b ≤ c dlq bud\-qkyx b, c S∈ .
Spoçatku pokaΩemo, wo ideal I1 = f TE∈{ rank (f) ≤ 1} [ wil\nym. Dlq ob-
©runtuvannq c\oho faktu dovedemo nyzku lem.
Lema 6. Nexaj E � napivreßitka skinçenno] dovΩyny, koΩnyj nenul\ovyj
element qko] [ ob�[dnannqm atomiv. Qkwo ϕ ∈TE takyj, wo R1( )ϕ � E TE( ),
to ϕ ∈ E TE( ).
Dovedennq provedemo indukci[g za ranhom. Qkwo rank( )ϕ = 1, to, oçevyd-
no, wo ϕ ∈ E TE( ). Nexaj bud\-qkyj element α napivhrupy TE , ranh qkoho ≤ k
i R1( )α � E TE( ), [ idempotentom. Dovedemo, wo bud\-qkyj element ψ, ranh
qkoho k + 1 i R1( )ψ � E TE( ), teΩ [ idempotentom. Nexaj dom( )ψ = bE. Os-
kil\ky bud\-qkyj nenul\ovyj element napivreßitky E [ ob�[dnannqm atomiv i,
krim toho, rank( )b ≥ 2, to element b pokryva[ prynajmni dva riznyx elementy
(skaΩimo, x i y). Vyberemo bud\-qkyj element z takyj, wo z b< . Todi
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STRUKTURA NAPIVHRUPY MANNA SKINÇENNOHO RANHU … 57
rank( )∆z ≤ k, a otΩe, rank( )∆z o ψ ≤ k. Oskil\ky ∆z o ψ � ψ , to R z1(∆ °
° ψ) � R1( )ψ � E TE( ). Zvidsy, za induktyvnym prypuwennqm, ∆z o ψ [ idempo-
tentom. OtΩe, zψ = z. Dovedemo teper, wo bψ = b. Prypustymo protyleΩne,
tobto bψ ≠ b. Nexaj bψ = c. Cilkom oçevydno, wo vidnoßennq pokryttq zberi-
ha[t\sq pry izomorfizmi. OtΩe, oskil\ky x bp i y bp , to x cp i y cp .
Zvidsy vyplyva[, wo bc = x i vodnoças bc = y. Supereçnist\. Takym çynom,
bψ = b. OtΩe, ψ � idempotent.
Lemu 6 dovedeno.
Dali, nexaj P � napivreßitka skinçenno] dovΩyny. Dlq elementa b ≠ 0
poznaçymo çerez A b( ) mnoΩynu x P∈{ x ≤ b ∧ rank (x) = 1} .
Lema 7. Nexaj P � napivreßitka skinçenno] dovΩyny, bud\-qkyj nenul\o-
vyj element qko] [ ob�[dnannqm atomiv. Todi dlq bud\-qkoho elementa b ≠ 0
sup ( )A b = b.
Dovedennq. Oskil\ky za umovog bud\-qkyj element napivreßitky P [
ob�[dnannqm atomiv, to b = sup B, de B � deqka pidmnoΩyna mnoΩyny atomiv.
Oçevydno, wo B � A b( ). Nexaj element c takyj, wo dlq bud\-qkoho x ∈ A b( )
vykonu[t\sq nerivnist\ x c≤ , todi c � verxnq meΩa mnoΩyny B. Oskil\ky
b = sup B, to b c≤ . OtΩe, b = sup ( )A b .
Lemu 7 dovedeno.
Lema 8. Nexaj E � napivreßitka skinçenno] dovΩyny, koΩnyj nenul\ovyj
element qko] [ ob�[dnannqm atomiv , TE � vidpovidna napivhrupa Manna. Ne-
xaj f, ϕ ∈TE taki, wo R f1( ) = R1( )ϕ , todi dom( )f = dom (ϕ), im( )f = im (ϕ).
Dovedennq. Nexaj dom( )f = bS i dom (ϕ) = cS. PokaΩemo, wo A b( ) = A c( ) .
Nexaj a ∈ A b( ), todi a b≤ , zvidky a ∈ dom( )f . Nexaj a f = a1. Oskil\ky a �
atom, to a1 teΩ [ atomom, tomu
0
0 1
a
a
TE
∈ i
0
0 1
1
a
a
R f
∈ ( ). Oskil\ky
R f1( ) = R1( )ϕ , to
0
0 1
1
a
a
R
∈ ( )ϕ . Zvidsy
0
0 1
a
a
� ϕ, todi a ∈ dom (ϕ), tobto
a cE∈ , a otΩe, a c≤ . Krim toho, rank (a) = 1. OtΩe, a A c∈ ( ) . Takym çynom,
A b( ) � A c( ) . Analohiçno dovodymo, wo A c( ) � A b( ). Zvidsy A c( ) = A b( ). Todi
za lemog 7 b = c. OtΩe, bS = cS, tobto dom( )f = dom (ϕ). Analohiçno moΩna
dovesty, wo im( )f = im (ϕ).
Lemu 8 dovedeno.
Lema 9. Nexaj E � napivreßitka skinçenno] dovΩyny, koΩnyj nenul\ovyj
element qko] [ ob�[dnannqm atomiv, TE � vidpovidna napivhrupa Manna.
Qkwo f, ϕ ∈TE taki, wo R f1( ) = R1( )ϕ , to f o ϕ–1 � idempotent.
Dovedennq. Nexaj
0
0
1
2
a
a
� f o ϕ–1 , de a1 i a2 � atomy napivreßitky
E. Todi isnu[ atom a E∈ takyj, wo
a
a
1
∈ f i
a
a2
∈ ϕ–1, zvidky
a
a
2
∈ ϕ.
OtΩe,
0
0
1
1
a
a
R f
∈ ( ) i
0
0
2
1
a
a
R
∈ ( )ϕ . Oskil\ky R f1( ) = R1( )ϕ , to
0
0
1a
a
∈
∈ R1( )ϕ , tomu
0
0
1a
a
� ϕ. Pozaqk
0
0
2a
a
� ϕ, to z ostannix dvox vklgçen\
ma[mo a1 = a2. OtΩe, R f1
1( )–o ϕ � E TE( ), tomu za lemog 6 f o ϕ–1 �
idempotent.
Lemu 9 dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
58 V. D. DEREÇ
Lema 10. Nexaj S � dovil\na inversna napivhrupa. Qkwo dlq elementiv b
i c vykonugt\sq rivnosti bb–1 = cc–1, b b–1 = c c–1 i bc–1 ∈ E S( ) , to b = c.
Dovedennq. Za umovog bc–1 bc–1 = bc–1, tomu bc–1 bc–1c = bc–1c, zvidky
bc–1 bb–1b = bb–1b abo bc–1b = b. Z ostann\o] rivnosti ma[mo bc–1 bb–1 = bb–1.
Zvidsy (vraxovugçy rivnist\ bb–1 = cc–1) bc–1 cc–1 = cc–1, otΩe, bc–1 = cc–1,
tomu bc–1c = cc–1 c = c. Krim toho, bc–1c = bb–1 b = b. Zvidsy b = c.
Lemu 10 dovedeno.
Lema 11. Nexaj E � napivreßitka skinçenno] dovΩyny, bud\-qkyj nenul\o-
vyj element qko] [ ob�[dnannqm atomiv, TE � vidpovidna napivhrupa Manna.
Qkwo f, ϕ ∈TE taki, wo R f1( ) = R1( )ϕ , to f = ϕ.
Dovedennq. Za lemog 8 dom( )f = dom (ϕ) i im( )f = im (ϕ). Za lemog 9 f °
° ϕ–1 [ idempotentom. Todi za lemog 10 f = ϕ.
Lemu 11 dovedeno.
Teper my moΩemo perejty do dovedennq implikaci] 2 → 1. OtΩe, nexaj E �
napivreßitka skinçenno] dovΩyny, koΩnyj nenul\ovyj element qko] [ ob�[dnan-
nqm atomiv. Nexaj Ω � stabil\nyj porqdok na napivhrupi TE , vidminnyj vid
rivnosti. Vraxovugçy rezul\tat (dyv. [6, s. 303]), nam slid dovesty, wo Ω ⊆ ω
abo Ω ⊆ ω–1 (de ω � kanoniçnyj porqdok na TE). Oskil\ky binarne vidno-
ßennq Ω vidminne vid rivnosti, to isnugt\ α, λ ∈TE taki, wo ( , )α λ ∈ Ω i α ≠
≠ λ. V statti [9] u zahal\nij formi dovedeno, wo funkciq F : σ � R1( )σ (de
σ ∈TE ) � homomorfizm iz napivhrupy TE v nadnapivhrupu P I( )1 , tobto napiv-
hrupu vsix neporoΩnix pidmnoΩyn idealu I1 = f TE∈{ rank( )f ≤ 1} vidnosno
zvyçajnoho hlobal\noho mnoΩennq. Za lemog 11 cej homomorfizm [ in�[ktyv-
nym, a otΩe (dyv. rezul\tat 3 u v cij statti), ideal I1 [ wil\nym. Z rezul\tatu 2
vyplyva[, wo ideal I1 [ reduktyvnym, a otΩe, zhidno z rezul\tatom 1 pravo- i li-
voreduktyvnym. Oskil\ky α ≠ λ, to isnu[ β ∈ I1 takyj, wo β αo ≠ β λo . Oçe-
vydno, wo ideal I1 [ napivhrupog Brandta z tryvial\nog strukturnog hrupog,
tomu za lemog 1 moΩlyvi lyße dva vypadky:
1) β αo =
0
0
i rank( )β λo = 1;
2) rank( )β αo = 1 i β λo =
0
0
.
Nexaj dlq konkretnosti ma[ misce perßyj vypadok, todi
0
0
∈, β λo Ω . (1)
Dovedemo, wo v c\omu vypadku Ω ⊆ ω (de ω � kanoniçnyj porqdok na invers-
nij napivhrupi TE). OtΩe, nexaj ( , )ϕ η ∈ Ω. Nam treba dovesty, wo ϕ η⊆ ,
abo, vraxovugçy rezul\tat 4, potribno pokazaty, wo R1( )ϕ � R1( )η . Prypusty-
mo protyleΩne, tobto isnu[ element ψ ∈ R1( )ϕ i ψ ∉ R1( )η . Todi ψ ≠
0
0
, a
otΩe, rank( )ψ = 1. Oskil\ky ψ ∈ R1( )ϕ , to ψ � ϕ. Zvidsy ψ ψo –1 o ϕ = ψ.
Pozaqk ( , )ϕ η ∈ Ω, to ψ( ° ψ–1 ° ϕ, ψ ° ψ–1 ° η) ∈ Ω. Tobto ψ( , ψ ° ψ–1 ° η) ∈
∈ Ω. Zvidsy
ψ ψ ψ η ψ ψ, – –o o o o1 1( ) ∈Ω . (2)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
STRUKTURA NAPIVHRUPY MANNA SKINÇENNOHO RANHU … 59
Rozhlqnemo element ψ ° ψ–1 ° η ° ψ–1 ° ψ. MoΩlyvi dva vypadky:
1) ψ ° ψ–1 ° η ° ψ–1 ° ψ =
0
0
;
2) rank ψ( ° ψ–1 ° η ° ψ–1 ° ψ) = 1.
Qkwo ψ ° ψ–1 ° η ° ψ–1 ° ψ =
0
0
, to ψ,
0
0
∈ Ω, pryçomu, qk vΩe zaznaça-
losq vywe, ψ ≠
0
0
. Oskil\ky ideal I1 [ 0-prostog napivhrupog, to I1 ° ψ °
° I1 = I1. OtΩe, znajdut\sq elementy τ, ξ ∈TE taki, wo β λo = τ ° ψ ° ξ. Takym
çynom, β λo ,
0
0
∈ Ω i
0
0
, β λo ∈ Ω (dyv. spivvidnoßennq (1)). Zvidsy
β λo =
0
0
. Supereçnist\.
Nexaj teper rank ψ( ° ψ–1 ° η ° ψ–1 o ψ) = 1. Z lemy 1 (vraxovugçy spivvid-
noßennq (2)) vyplyva[ ψ ° ψ–1 ° η ° ψ–1 ° ψ = ψ. Ale ψ ° ψ–1 ° η ° ψ–1 ° ψ �
� η, tobto ψ η⊆ . OtΩe, ψ ∈ R1( )η . Supereçnist\. Takym çynom, R1( )ϕ �
� R1( )η . Zvidsy (dyv. rezul\tat 4) ϕ η⊆ , tobto ( , )ϕ η ∈ ω.
OtΩe, my dovely ekvivalentnist\ umov 1 i 2 (dyv. teoremu 1).
Implikaciq 2 → 3 bezposeredn\o vyplyva[ z lemy 11 i rezul\tatu 3. Teper ob-
©runtu[mo implikacig 3 → 2. Dlq c\oho dovedemo we odne tverdΩennq.
Lema 12. Nexaj S � inversna napivhrupa skinçenno] dovΩyny i idempotenty
b ta c taki, wo R b1( ) = R c1( ). Todi dlq bud\-qkoho x, wo naleΩyt\ idealu
I1 = z S∈{ rank( )z ≤ 1} , ma[ misce rivnist\ x b = x c.
Dovedennq. Nexaj x I∈ 1, todi, oçevydno, x–1x b ∈ R b1( ) . Oskil\ky za umo-
vog R b1( ) = R c1( ), to x–1x b ∈ R c1( ). Zvidsy x–1x bc = x–1x b. OtΩe,
x b = xx xb–1 = xx xbc–1 = xbc . (3)
Analohiçno, oskil\ky x–1x c ∈ R c1( ) i R b1( ) = R c1( ), to x–1x c ∈ R b1( ) . Zvidsy
x–1x cb = x–1x c. OtΩe,
x c = xx xc–1 = xx xcb–1 = xbc . (4)
Z (3) i (4) vyplyva[, wo x b = x c.
Lemu 12 dovedeno.
Teper implikacig 3 → 2 dovedemo vid suprotyvnoho. Prypustymo, wo v na-
pivreßitci [ nenul\ovyj element, qkyj ne moΩna podaty qk ob�[dnannq atomiv.
Todi za lemog 3 isnu[ element (poznaçymo joho çerez c) takyj, wo pokryva[
toçno odyn nenul\ovyj element (skaΩimo, b), do toho Ω rank( )c ≥ 2. Rozhlqne-
mo ∆b i ∆c , qki naleΩat\ E TE( ). Oskil\ky napivreßitka E izomorfna napiv-
reßitci E TE( ), to ∆c pokryva[ toçno odyn idempotent , a same, ∆b . Z lemy 4
vyplyva[, wo R b1( )∆ = R c1( )∆ . Za lemog 12 robymo vysnovok, wo dlq bud\-
qkoho α ∈ I1 = f S∈{ rank( )f ≤ 1} ma[ misce rivnist\ α o ∆b = α o ∆c . Ale Ω
∆b ≠ ∆c . Takym çynom, ideal I1 ne [ livoreduktyvnym, a otΩe (dyv. rezul\taty
1 i 2), ideal I1 ne [ wil\nym. Supereçnist\.
Implikacig 3 → 2 dovedeno.
Dali, napivhrupu S nazyvagt\ perestavnog, qkwo bud\-qki dvi ]] konhruenci]
komutugt\ vidnosno zvyçajno] operaci] superpozyci] binarnyx vidnoßen\. Qk i
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
60 V. D. DEREÇ
raniße, çerez E poznaçymo napivreßitku skinçenno] dovΩyny (dovΩyna ne men-
ßa 2). U statti [10, s. 746] dovedeno, wo koΩnyj nenul\ovyj ideal perestavno]
napivhrupy Manna TE [ wil\nym. Zvidsy oderΩu[mo naslidky teoremy 1.
Naslidok 1. KoΩnyj stabil\nyj porqdok na perestavnij napivhrupi Manna [
fundamental\nym abo antyfundamental\nym.
Naslidok 2. KoΩnyj nenul\ovyj idempotent perestavno] napivhrupy Manna
[ ob�[dnannqm atomiv.
1. Munn W. D. Fundamental inverse semigroups // Quart. J. Math. Oxford. – 1970. – 21. – P. 157 –
170.
2. Petrich M. Inverse semigroups. – New York etc.: John Willey and Sons, 1984. – 674 p.
3. Dereç V. D. Konhruenci] perestavno] inversno] napivhrupy skinçennoho ranhu // Ukr. mat.
Ωurn. � 2005. � 57, # 4. � S. 469 � 473.
4. Dereç V. D. Pro perestavni konhruenci] na antyhrupax skinçennoho ranhu // Tam Ωe. � 2004.
� 56, # 3. � S. 346 � 351.
5. Íajn B. M. Predstavlenye uporqdoçenn¥x poluhrupp // Mat. sb. � 1964. � 65, # 2. � S. 188
� 197.
6. Goberstein S. M. Fundamental order relations on inverse semigroups and on their generalizations //
Semigroup Forum. – 1980. – 21. – P. 285 – 328.
7. Klyfford A., Preston H. Alhebrayçeskaq teoryq poluhrupp: V 2 t. � M.: Myr, 1972. �
T. 1, 2.
8. Schein B. M. Completions, translational hulls and ideal extensions of inverse semigroups // Czech.
Math. J. – 1973. – 98. – P. 575 – 610.
9. Dereç V. D. Pro maksymal\ni stabil\ni porqdky na inversnij napivhrupi skinçennoho ranhu z
nulem // Ukr. mat. Ωurn. � 2008. � 60, # 8. � S. 1035 � 1041.
10. Dereç V. D. Struktura perestavno] napivhrupy Manna skinçennoho ranhu // Tam Ωe. � 2006. �
58, # 6. � S. 742 � 746.
OderΩano 29.01.08
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|
| id | umjimathkievua-article-3000 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:21Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e5/84a8e62a2b9a7b9f8415f7b9bba0e2e5.pdf |
| spelling | umjimathkievua-article-30002020-03-18T19:43:07Z Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental Структура напівгрупи Манна скінченного рангу, кожний стабільний порядок якої є фундаментальним або антифундаментальним Derech, V. D. Дереч, В. Д. We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental. Описывается структура полугруппы Манна конечного ранга, каждый стабильный порядок которой фундаментальный или антифундаментальный. Institute of Mathematics, NAS of Ukraine 2009-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3000 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 1 (2009); 52-60 Український математичний журнал; Том 61 № 1 (2009); 52-60 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3000/2749 https://umj.imath.kiev.ua/index.php/umj/article/view/3000/2750 Copyright (c) 2009 Derech V. D. |
| spellingShingle | Derech, V. D. Дереч, В. Д. Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental |
| title | Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental |
| title_alt | Структура напівгрупи Манна скінченного рангу, кожний стабільний порядок якої є фундаментальним або антифундаментальним |
| title_full | Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental |
| title_fullStr | Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental |
| title_full_unstemmed | Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental |
| title_short | Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental |
| title_sort | structure of a munn semigroup of finite rank every stable order of which is fundamental or antifundamental |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3000 |
| work_keys_str_mv | AT derechvd structureofamunnsemigroupoffiniterankeverystableorderofwhichisfundamentalorantifundamental AT derečvd structureofamunnsemigroupoffiniterankeverystableorderofwhichisfundamentalorantifundamental AT derechvd strukturanapívgrupimannaskínčennogorangukožnijstabílʹnijporâdokâkoíêfundamentalʹnimaboantifundamentalʹnim AT derečvd strukturanapívgrupimannaskínčennogorangukožnijstabílʹnijporâdokâkoíêfundamentalʹnimaboantifundamentalʹnim |