Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero
We give a characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero.
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| Date: | 2007 |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509476702912512 |
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| 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 |
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| datestamp_date | 2020-03-18T19:53:10Z |
| description | We give a characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero. |
| first_indexed | 2026-03-24T02:41:43Z |
| format | Article |
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UDK 512.534.5
V. D. Dereç (Vinnyc. nac. texn. un-t)
XARAKTERYSTYKA NAPIVREÍITKY IDEMPOTENTIV
PERESTAVNO} INVERSNO} NAPIVHRUPY
SKINÇENNOHO RANHU Z NULEM
The characterization of a semilattice of idempotents of a finite-rank permutable inverse semigroup with
zero is given.
Dana xarakterystyka polureßetky ydempotentov perestanovoçnoj ynversnoj poluhrupp¥ ko-
neçnoho ranha s nulem.
Vstup. Napivhrupa nazyva[t\sq perestavnog, qkwo bud\-qki dvi ]] konhruenci]
komutugt\ vidnosno zvyçajno] operaci] superpozyci] binarnyx vidnoßen\. Do
klasu perestavnyx napivhrup naleΩat\ hrupy, napivhrupy Brandta ta inßi. Za-
znaçymo, wo napivhrupy, konhruenci] qkyx utvorggt\ lancgΩok vidnosno
vklgçennq (ce tak zvani ∆-napivhrupy), oçevydno, takoΩ [ perestavnymy. Qk
vidomo (dyv. [1, t. 2, s. 287] ), bud\-qka symetryçna skinçenna napivhrupa [ ∆-na-
pivhrupog. Komutatyvni ∆-napivhrupy nezaleΩno opysani T.5Tamurog [2] i
B.5Íajnom [3, 4]. V bil\ßosti nastupnyx robit (vidpovidnu bibliohrafig moΩna
znajty v ohlqdovyx stattqx [5, 6], a takoΩ v roboti [7] ) z’qsovu[t\sq struktura
∆-napivhrup, qki [ uzahal\nennqm komutatyvnyx napivhrup. U statti [8] zapropo-
novano konstrukcig skinçenno] inversno] ∆-napivhrupy.
Teper wodo perestavnyx napivhrup, qki ne obov’qzkovo [ ∆-napivhrupamy. V
statti [9] vstanovleno strukturu komutatyvno] perestavno] napivhrupy. V po-
dal\ßyx robotax vyvçalysq perestavni napivhrupy, wo blyz\ki do komutatyvnyx
(napryklad, duo-napivhrupy [10], medial\ni napivhrupy [11], LC -komutatyvni
napivhrupy [12], RDGCn-komutatyvni napivhrupy [13] ta in.), a takoΩ perestavni
cilkom rehulqrni napivhrupy (dyv. [6], teoremy54.8 i54.9) i rehulqrni ω-napiv-
hrupy [14] .
Perestavni inversni napivhrupy rozhlqdalysq v kil\kox robotax. Krim vywe-
zhadano] statti [8] (qka, vtim, stosu[t\sq inversnyx ∆-napivhrup) vidomog [ teo-
rema (dyv. [5], teorema56.3), v qkij znajdeno neobxidni i dostatni umovy dlq toho,
wob skinçenna inversna napivhrupa bula perestavnog. Krim toho, z rezul\tativ
Tamury i Hamil\tona bezposeredn\o vyplyva[ teorema pro budovu perestavno]
klifordovo] napivhrupy [6, c. 34]. Neobxidni i dostatni umovy dlq toho, wob in-
versna ω-napivhrupa bula perestavnog, znajdeno v roboti [14]. U statti [15]
z’qsovu[t\sq struktura bud\-qko] konhruenci] perestavno] antyhrupy vsix izo-
morfizmiv miΩ holovnymy idealamy napivreßitky skinçenno] dovΩyny. V statti
[16] cej rezul\tat uzahal\ng[t\sq — znajdeno strukturu bud\-qko] konhruenci]
perestavno] inversno] napivhrupy skinçenno] dovΩyny z nulem. U danij roboti
znajdeno xarakterystyku napivreßitky idempotentiv perestavno] inversno]
napivhrupy skinçennoho ranhu z nulem. Osnovnymy rezul\tatamy statti [ teo-
remy52 i55.
1. Osnovna terminolohiq i poznaçennq. Napivreßitka P nazyva[t\sq na-
pivreßitkog skinçenno] dovΩyny, qkwo isnu[ natural\ne çyslo n take, wo dov-
Ωyna bud\-qkoho lancgΩka z S ne perevywu[ çysla n.
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 ) min rank( ), rank( )(a b a b⋅ ≤ { }.
Çyslo rank( )a nazyva[t\sq ranhom elementa a.
© V. D. DEREÇ, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10 1353
1354 V. D. DEREÇ
Nexaj S — inversna napivhrupa, napivreßitka idempotentiv qko] ma[ skinçen-
nu dovΩynu. Funkciq rank( )a = h a a( )⋅ −1
, de h a a( )⋅ −1
— vysota idempotenta
a a⋅ −1
v napivreßitci idempotentiv napivhrupy S, [ ranhovog funkci[g (dyv. [16,
c. 470]). U danij roboti my operu[mo ranhom elementa same v takomu rozuminni.
Vsi inßi neobxidni ponqttq z teori] napivhrup moΩna znajty v monohrafi] [1].
2. Pro perestavnu inversnu napivhrupu z nulem. Spoçatku rozhlqnemo
najprostißyj vypadok, a same, koly ranh bud\-qkoho elementa inversno] napiv-
hrupy ne perevywu[ odynycg.
Teorema 1. Nexaj S — inversna napivhrupa z nulem taka, wo dlq bud\-qkoho
x rank ( x ) ≤ 1.
Todi S [ perestavnog v tomu i lyße v tomu vypadku, koly vona [ napiv-
hrupog Brandta.
Dovedennq. Nexaj napivhrupa S [ perestavnog, todi za teoremog54 (dyv.
[9]) ]] idealy linijno vporqdkovani, a otΩe (dyv. teoremu52 z [16]), koΩnyj ]] ide-
al ma[ formu { x S x∈ rank( ) ≤ k }. V danomu vypadku k = 0 abo k = 1. Lehko
zrozumity, wo mnoΩyna { x S x∈ rank( ) = 0 } [ odnoelementnog ([dynym ]] ele-
mentom [ nul\). Takym çynom, napivhrupa S [ 0-prostog. Krim toho, bud\-qkyj
idempotent, ranh qkoho dorivng[ 1, [ prymityvnym. OtΩe, S [ cilkom 0-
prostog inversnog napivhrupog, tobto napivhrupog Brandta.
Obernene tverdΩennq oçevydnym çynom vyplyva[ z vidomoho rezul\tatu
(dyv., napryklad, [1]) pro strukturu bud\-qko] konhruenci] napivhrupy Brandta.
Dali budemo vvaΩaty, wo inversna napivhrupa S mistyt\ element, ranh qkoho
ne menßyj niΩ 2.
Sformulg[mo perßyj (sered dvox osnovnyx) rezul\tativ statti (vin anonso-
vanyj u [17]).
Teorema 2. Nexaj S — inversna napivhrupa z nulem, napivreßitka E idem-
potentiv qko] ma[ skinçennu dovΩynu.
Todi S [ perestavnog v tomu i lyße v tomu vypadku, koly vykonugt\sq
taki dvi umovy:
1) dlq bud\-qkyx a i b ∈ S, qkwo rank( )a = rank( )b , to SaS = SbS ;
2) dlq bud\-qkoho e ∈ E ( )rank( ) 2e ≥ isnugt\ idempotenty f i ω taki,
wo f ≠ ω, f < e, ω < e i rank( )f = rank( )ω = rank( )e – 1.
Pered tym, qk bezposeredn\o perejty do dovedennq teoremy, nam potribno
dovesty kil\ka lem.
OtΩe, nexaj L — napivreßitka skinçenno] dovΩyny. Oçevydno, vona mistyt\
nul\.
Budemo hovoryty, wo L zadovol\nq[ umovu D , qkwo 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 ∈ L ( rank( )e ≥ 2) isnugt\ elementy b , c ∈ L taki, wo
b ≠ c, b < e, c < e i rank( )b = rank( )c = rank( )e – 1.
Lema 1. Dlq napivreßitky L skinçenno] dovΩyny umovy D i R [ ekviva-
lentnymy.
Dovedennq. Prypustymo, wo vykonu[t\sq umova D . Nexaj e ∈ L, pryçomu
rank( )e ≥ 2. Zrozumilo, wo isnu[ element b takyj, wo b < e i rank( )b =
=5 rank( )e – 1. Za umovog D isnu[ element c takyj, wo c ≠ b, c < e i rank( )c =
=5 rank( )b . OtΩe, vykonu[t\sq umova R.
Navpaky, nexaj vykonu[t\sq umova R. Prypustymo, wo a < b i rank( )a ≥ 1.
Vypadok A. Element a naleΩyt\ maksymal\nomu (za kil\kistg elemen-
tiv) lancgΩku, wo z’[dnu[ 0 i b.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
XARAKTERYSTYKA NAPIVREÍITKY IDEMPOTENTIV PERESTAVNO} INVERSNO} .… 1355
V c\omu lancgΩku beremo element s takyj, wo a � s (tut � poznaça[ po-
kryttq). Todi rank( )s = rank( )a + 1. Za umovog R isnugt\ elementy x i y ta-
ki, 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 =
=5 rank( )a = rank( )s – 1. OtΩe, vykonu[t\sq umova D.
Vypadok B. Element a ne naleΩyt\ Ωodnomu maksymal\nomu (za kil\kis-
tg elementiv) lancgΩku, wo z’[dnu[ 0 i b.
Nexaj P — takyj maksymal\nyj lancgΩok v L. Isnu[ element u ∈ P ta-
kyj, wo rank( )u = rank( )a + 1. Za umovog R znajdut\sq elementy z i v taki,
wo z ≠ v, z < u, v < u, rank( )z = rank( )v = rank( )u – 1 = rank( )a . Zrozumilo, wo
z ≠ a abo v ≠ a. Nexaj, napryklad, z ≠ a. Krim toho, z < b i rank( )z =
=5 rank( )a . OtΩe, vykonu[t\sq umova D.
Dali umovog 2 (dyv. teoremu52) my budemo korystuvatysq abo v formi R abo
v formi D.
Lema 2. Nexaj S — inversna napivhrupa, napivreßitka idempotentiv E
qko] ma[ skinçennu dovΩynu. Nexaj idempotenty a i b taki, wo a = xbx−1
dlq deqkoho x ∈ S, pryçomu rank( )b = rank( )a = rank( )x .
Todi aE ≅ bE.
Dovedennq. Spoçatku vidmitymo spravedlyvist\ rivnosti
xx a− =1
. (1)
Dijsno, oskil\ky a = xbx−1
, to xx a−1
= xx xbx− −1 1
= xbx−1
= a. OtΩe, a ≤
≤ xx−1
. Qkwo prypustyty, wo a < xx−1
, to rank( )a < rank( )xx−1
= rank( )x , wo
supereçyt\ umovi. OtΩe, xx−1
= a.
Teper pokaΩemo, wo
ax xb= . (2)
Dijsno, ax = xbx x−1 = xx xb−1 = xb. Dali pokaΩemo, wo
rank( ) = rank( )xb b . (3)
Prypustymo protyleΩne, tobto rank( )xb < rank( )b . Ma[mo rank( )a =
= rank( )xbx−1 ≤ rank( )xb < rank( )b , wo supereçyt\ umovi.
Teper ob©runtu[mo rivnist\
x x b− =1
. (4)
Dlq c\oho rozhlqnemo vsi moΩlyvi vypadky:
a) x x−1 < b,
b) b < x x−1
,
c) idempotenty x x−1
i b utvorggt\ antylancgΩok,
d) x x−1 = b.
Qkwo prypustyty, wo x x−1 < b , to rank( )x = rank( )x x−1 < rank( )b , wo
supereçyt\ umovi. Tobto vypadok a) nemoΩlyvyj.
Analohiçno ne vykonu[t\sq umova b). Dali, prypustymo, wo vykonu[t\sq
umova5s), todi x xb−1 < x x−1
. Skorystavßys\ rivnistg (3), oderΩymo
rank( )x xb−1 = rank( )xb = rank( )b . Ale rank( )x xb−1 < rank( )x x−1
. Zvidsy
rank( )b < rank( )x x−1 = rank( )x , wo supereçyt\ umovi.
Takym çynom, zalyßa[t\sq odna moΩlyvist\: x x−1 = b.
PokaΩemo, wo
x ax b− =1
. (5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1356 V. D. DEREÇ
Dijsno, zastosovugçy rivnosti (1) i (4), ma[mo x ax−1 = x xx x− −1 1 = x x−1 = b.
Teper dovedemo izomorfizm napivreßitok bE i aE. Vyznaçymo funkcig
F : bE → aE. A same, qkwo c ∈ bE, to F ( c ) = xcx−1
.
1. Dovedemo, wo F ( c ) ∈ aE. Dijsno, oskil\ky c ∈ bE , to c = cb. Dali,
F ( c ) = xcx−1 = xcbx−1
. Z rivnosti (2) vyplyva[ rivnist\ x a−1 = bx−1
, tomu
F ( c ) = xcx a−1
. OtΩe, xcx−1 ≤ a, tobto xcx−1
∈ aE.
2. Dovedemo, wo F — sgr’[ktyvna funkciq, tobto F ( bE ) = aE. Nexaj
m5∈ a E , todi m a = m. Spoçatku pokaΩemo, wo x mx−1
∈ bE . Dijsno,
x mx−1 = x−1max = [zastosovu[mo rivnist\ (2)] = x mxb−1
. OtΩe, x mx−1 ≤ b, tob-
to x mx−1
∈ bE . Dali, F x mx( )−1
= xx mxx− −1 1 = [zastosovu[mo5rivnist\5(1)] =
= ama = am = m.
3. Dovedemo, wo F — in’[kciq. Nexaj b1 ∈ bE i b2 ∈ bE, pryçomu b1 ≠ b2.
PokaΩemo, wo F ( b1 ) ≠ F ( b2 ). Prypustymo protyleΩne, tobto F ( b1 ) = F ( b2 )
abo xb x1
1− = xb x2
1− , todi x xb x x− −1
1
1
= x xb x x− −1
2
1 = [zastosovu[mo rivnist\
(4)] = bb b1 = bb b2 . Zvidsy b1 = b2. Supereçnist\.
Nareßti dovedemo, wo F — izomorfizm. Nexaj b1 ∈ bE i b2 ∈ bE. Todi
F b F b( ) ( )1 2⋅ = xb x xb x1
1
2
1− − = xx xb b x− −1
1 2
1 = xb b x1 2
1− = F b b( )1 2⋅ .
Lema 3. Nexaj S — inversna napivhrupa, napivreßitka idempotentiv qko] E
ma[ skinçennu dovΩynu.
Qkwo idempotenty a i b taki, wo SaS = SbS, to aE ≅ bE.
Dovedennq. Znajdut\sq elementy x i y taki, wo a = xby. Dovedemo, wo
a5= axb ax( )−1 = axbx a−1
. Dijsno, axbx a−1 = axbx xby−1 = axx xbby−1 = axby =
= aa = a. Z rivnosti SaS = SbS vyplyva[, wo rank( )a = rank( )b . Dali, rank( )a =
= rank( )axbx a−1 ≤ rank(ax) . Krim toho, rank(ax) ≤ rank( )a . OtΩe, rank( )a =
= rank( )b = rank(ax) . Takym çynom, za poperedn\og lemog aE ≅ bE.
Perejdemo do dovedennq teoremy52. Spoçatku dovedemo dostatnist\. Ot-
Ωe, nexaj vykonugt\sq umovy 1 i 2 teoremy52. Umova 1 zabezpeçu[ linijnu vpo-
rqdkovanist\ (vidnosno vklgçennq) idealiv napivhrupy S (dyv. [16], teoremu52).
PokaΩemo teper, wo bud\-qka konhruenciq Θ napivhrupy S ma[ formu Θ = I ×
× I ∪ Ω, de I — ideal napivhrupy S, a Ω ⊆ H (H — vidnoßennq Hrina). OtΩe,
nexaj Θ — konhruenciq na S . Lehko pereviryty, wo mnoΩyna IΘ 5=
=5 x S x∈ ∈{ },0 Θ [ idealom. Oskil\ky koΩnyj ideal napivhrupy S [ ranhovym
(dyv. [16], teoremu52), to isnu[ natural\ne çyslo k take, wo IΘ = Ik 5=
=5 x S x k∈ ≤{ }rank( ) . Nexaj 〈 x, y 〉 ∈ Θ i rank( )x > k, todi, oçevydno, i
rank( )y 5 > k.
PokaΩemo spoçatku, wo rank( )x = rank( )y . Prypustymo protyleΩne, tobto
rank( )x ≠ rank( )y . Nexaj dlq konkretnosti rank( )x < rank( )y . Oskil\ky
〈 x, y 〉5∈ Θ, to 〈 xx−1, yy−1
〉 ∈ Θ. Zvidsy 〈 xx yy− −1 1, yy−1〉 ∈ Θ.
Rozhlqnemo moΩlyvi vypadky.
Perßyj vypadok: rank( )1xx yy− −1 ≤ k.
Todi 〈 0, yy−1〉 ∈ Θ. Zvidsy yy−1
∈ Ik
, tobto rank( )yy−1 ≤ k . Ale
rank( )yy−1 = rank( )y > k. Supereçnist\.
Druhyj vypadok: rank( )xx yy− −1 1 > k.
Zrozumilo, wo xx yy− −1 1 ≤ yy−1
, a oskil\ky rank( )xx yy− −1 1 ≤ rank( )xx−1 =
= rank( )x < rank( )y = rank( )yy−1
, to xx yy− −1 1 < yy−1
. Za umovog 2 (tobto umo-
vog R, qka za lemog 1 ekvivalentna umovi D) isnu[ idempotent w ∈ E takyj, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
XARAKTERYSTYKA NAPIVREÍITKY IDEMPOTENTIV PERESTAVNO} INVERSNO} .… 1357
rank( )xx yy− −1 1 = rank( )w , w < yy−1
, w ≠ xx yy− −1 1
. Oskil\ky 〈 xx yy− −1 1, yy−1〉 ∈
∈ Θ, to 〈 xx yy w− −1 1 , yy w−1 〉 ∈ Θ abo 〈 xx w−1 , w〉 ∈ Θ.
Qkwo rank( )xx w−1 ≤ k, to 〈 0, w〉 ∈ Θ. Zvidsy rank ( )w ≤ k. Supereçnist\.
Qkwo Ω rank( )xx w−1 > k, to zastosu[mo do vporqdkovano] pary 〈 xx w−1 , w〉
taki sami mirkuvannq, qk i vywe. ProdovΩugçy cej proces (a vin, oçevydno,
skinçennyj), oderΩu[mo 〈 0, b 〉 ∈ Θ dlq deqkoho b ∈ E, pryçomu rank( )b > k.
Supereçnist\. Takym çynom, rank( )x = rank( )y .
Teper pokaΩemo, wo xx−1 = yy−1
i x x−1 = y y−1
, tobto 〈x, y〉 ∈ H, de H [
vidnoßennqm Hrina. Oskil\ky 〈 x, y 〉 ∈ Θ, to 〈 x xx y, −1 〉 ∈ Θ. Oçevydno, ma[ mis-
ce nerivnist\ xx y−1 ≤ y. Qkwo prypustyty, wo xx y−1 < y , to rank( )xx y−1
<
< rank( )y . Z inßoho boku, 〈y, xx y−1 〉 ∈ Θ, tomu rank ( )xx y−1 = rank( )y . Supe-
reçnist\. Takym çynom, xx y−1 = y. Zvidsy xx yy− −1 1 = yy−1
. OtΩe, yy−1 ≤ xx−1
.
Qkwo prypustyty, wo yy−1
< xx−1, to rank( )y < rank( )x . Supereçnist\. Ot-
Ωe, yy−1 = xx−1. Analohiçno dovodyt\sq, wo y y−1 = x x−1
. OtΩe, idealy napiv-
hrupy S linijno vporqdkovani i bud\-qka konhruenciq Θ ma[ formu Θ = I × I ∪
∪ Ω (de I — ideal, a Ω ⊆ H ). Takym çynom, za teoremog54 (dyv. [16]) napiv-
hrupa S [ perestavnog.
Teper dovedemo neobxidnist\. Nexaj napivhrupa S [ perestavnog, todi (dyv.
[9], teoremu54) ]] idealy utvorggt\ lancgΩok vidnosno vklgçennq, a otΩe (dyv.
[16], teoremu52), vykonu[t\sq umova 1 teoremy.
Dovedemo, wo vykonu[t\sq j umova 2. Dovedennq provedemo vid suprotyvno-
ho, tobto prypustymo, wo isnu[ idempotent w ∈ S, dlq qkoho umova 2 (dyv. for-
mulgvannq teoremy52) ne vykonu[t\sq. Nexaj rank( )w = k + 1, de k + 1 ≥ 2.
Rozhlqnemo na S binarne vidnoßennq Σ = Ik−1 × Ik−1 ∪ ρ ∪ ∆, de ρ = {〈 x, y 〉 x <
< y ∧ rank( )x = k ∧ rank( )y = k + 1}, ∆ = {〈 x, x 〉 x ∈ S}, Ik−1 = {x ∈ S
rank( )x ≤ k – 1}. Dovedemo dvostoronng stabil\nist\ binarnoho vidnoßennq Σ.
Dovedennq rozib’[mo na kil\ka lem.
Lema 4. Nexaj 〈 a, b 〉 ∈ ρ (tobto a ≤ b, rank( )a = k i rank( )b = k + 1).
Qkwo dlq deqkoho c ∈ S rank( )ac = k, to 〈 ac, bc 〉 ∈ Σ. Qkwo Ω rank ( )ca =
= k, to 〈 ca, cb 〉 ∈ Σ.
Dovedennq. Oskil\ky za umovog a < b , to ac ≤ bc. Qkwo ac = bc, to
〈 ac, bc 〉 ∈ Σ. Qkwo Ω ac < bc, to k = rank( )ac < rank( )bc ≤ rank( )b = k + 1 .
OtΩe, rank( )bc = k + 1. Takym çynom, 〈 ac, bc 〉 ∈ ρ ⊂ Σ.
Druha çastyna lemy dovodyt\sq analohiçno.
Lema 5. Qkwo rank( )a = rank( )ab = m, to a a−1 ≤ bb−1
. Qkwo Ω
rank( )a = rank( )ba , to aa−1 ≤ b b−1
.
Dovedennq. Dovedemo perßu polovynu lemy (druha dovodyt\sq analohiç-
no). MoΩlyvi try vypadky:
a) bb−1 < a a−1
;
b) elementy a a−1
i bb−1
utvorggt\ antylancgΩok;
c) a a−1 ≤ bb−1
.
Prypustymo, wo bb−1 < a a−1
, todi m = rank( )ab = rank( )abb−1 ≤
≤ rank( )bb−1 < rank ( )a a−1 = rank ( )a = m. Supereçnist\.
Nexaj teper elementy bb−1
i a a−1
utvorggt\ antylancgΩok. Todi
a abb− −1 1 < a a−1
. PokaΩemo, wo rank( )a abb− −1 1 = m. Dijsno, m = rank( )ab =
= rank( )aa abb b− −1 1 ≤ rank( )a abb− −1 1
. Z inßoho boku, rank( )a abb− −1 1 ≤ m. Ot-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1358 V. D. DEREÇ
Ωe, rank( )a abb− −1 1 = m. Dali, oskil\ky a abb− −1 1 < a a−1
, to m =
= rank( )a abb− −1 1 < rank( )a a−1 = rank( )a = m. Supereçnist\.
Zalyßa[t\sq odna moΩlyvist\ : a a−1 ≤ bb−1
.
Lema 6. Nexaj a < b, krim toho, rank( )a = k i rank( )b = k + 1.
Qkwo rank( )bc = k + 1, to rank( )ac = k. Qkwo Ω rank( )cb = k + 1, to
rank( )ca = k.
Dovedennq. Dovedemo perßu çastynu lemy (druha dovodyt\sq analohiçno).
Oskil\ky rank( )bc = rank( )b , to za lemog55 b b−1 ≤ cc−1
. Zvidsy
a ab b− −1 1 ≤ a acc− −1 1
. Ale a ab b− −1 1 = a a−1
, tomu a a−1 ≤ a acc− −1 1
. OtΩe, k =
= rank( )a a−1 ≤ rank( )a acc− −1 1 ≤ rank( )ac . Z inßoho boku, rank( )ac ≤ rank( )a =
= k. Takym çynom, rank( )ac = k.
Dali budemo dovodyty pravostoronng stabil\nist\ binarnoho vidnoßennq Σ.
Nexaj 〈 a, b 〉 ∈ Σ. Qkwo 〈 a, b 〉 ∈ Ik−1 × Ik−1 abo a = b, to dovodyty nema[ çoho.
Rozhlqnemo vypadok, koly 〈 a, b 〉 ∈ ρ, tobto a < b, rank( )a = k i rank( )b =
= k + 1.
Nexaj c ∈ Σ. Rozhlqnemo moΩlyvi vypadky:
A) rank( )ac = k .
Todi za lemog 4 〈 ac, bc 〉 ∈ ρ.
V) rank( )ac < k .
Todi za lemog 6 rank( )bc ≤ k. PokaΩemo, wo rank( )bc < k. Prypustymo, wo
rank( )bc = k. Rozhlqnemo idempotenty aa−1
, bb−1
i bcc b− −1 1
. Zrozumilo, wo
magt\ misce nerivnosti aa−1 < bb−1
, bcc b− −1 1 < bb−1
. Krim toho, rank( )aa−1
=
= rank( )bcc b− −1 1 = k i rank( )bb−1 = k + 1. Oskil\ky rank( )w = rank( )bb−1
, to
(dyv. [16], teoremu52) SwS = Sbb S−1
. OtΩe, za lemog53 wE ≅ bb E−1
(tut ≅ po-
znaça[ izomorfizm). Zvidsy robymo vysnovok, wo aa−1 = bcc b− −1 1
. Dali, oskil\-
ky a < b, to a = ab b−1
i ab−1 = aa−1
. Zvidsy ab b−1 = aa b−1 = a. PomnoΩyvßy
rivnist\ aa−1 = bcc b− −1 1
zliva na aa−1
, oderΩymo aa−1 = aa bcc b− − −1 1 1 =
= acc b− −1 1
. OtΩe, k = rank( )aa−1 = rank( )acc b− −1 1 ≤ rank( )ac < k. Supereçnist\.
Takym çynom, rank( )bc < k. Zvidsy 〈 ac, bc 〉 ∈ Ik−1 × Ik−1 ⊆ Σ. OtΩe, binarne
vidnoßennq Σ = Ik−1 × Ik−1 ∪ ρ ∪ ∆ [ stabil\nym sprava. Analohiçno dovodyt\-
sq stabil\nist\ zliva. Lehko zrozumity, wo binarne vidnoßennq Σ = Ik−1 ×
×5 Ik−15 ∪ ρ−1
∪ ∆ teΩ stabil\ne, a tomu i binarne vidnoßennq Θ = Ik−1 × Ik−1 ∪
∪ ρ ∪ ρ−1 ∪ ∆ teΩ [ stabil\nym. Dali, poznaçymo çerez Θt
tranzytyvne
zamykannq binarnoho vidnoßennq Θ. Lehko pereviryty, wo Θt
— konhruenciq.
Z [16] (teorema54) vidomo, wo inversna napivhrupa S skinçennoho ranhu z nulem [
perestavnog todi i til\ky todi, koly koΩna konhruenciq ma[ formu I × I ∪ Ω,
de I — ideal napivhrupy S, a Ω ⊆ H (H — vidnoßennq Hrina). Konhruenciq
Θt
, qku my skonstrugvaly, vyxodqçy z prypuwennq, wo umova52 (dyv. formulg-
vannq teoremy52) ne vykonu[t\sq, ne pidpada[ pid navedenu vywe formu konhru-
enci] perestavno] inversno] napivhrupy skinçennoho ranhu z nulem. Dijsno,
Ik−15= { x ∈ S 〈x, 0〉 ∈ Θt} i, krim toho, isnu[ para 〈 a, b 〉 ∈ Θt
taka, wo
rank( )a = k, rank( )b = k + 1, tobto 〈 a, b 〉 ∉ H.
OderΩana supereçnist\ i zaverßu[ dovedennq teoremy52.
Naslidok. Nexaj S — perestavna inversna napivhrupa z nulem, napivreßit-
ka idempotentiv qko] ma[ skinçennu dovΩynu.
Bud\-qkyj ideal napivhrupy S [ perestavnog napivhrupog.
ZauvaΩennq. Cej naslidok takoΩ bezposeredn\o vyplyva[ z lemy 3.2
statti5[8].
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
XARAKTERYSTYKA NAPIVREÍITKY IDEMPOTENTIV PERESTAVNO} INVERSNO} .… 1359
3. Struktura perestavno] inversno] napivhrupy skinçennoho ranhu bez
nulq. V c\omu punkti my z’qsu[mo strukturu perestavno] inversno] napivhrupy
skinçennoho ranhu, wo ne mistyt\ nulq.
Ma[ misce takyj rezul\tat.
Teorema 3. Nexaj S — inversna napivhrupa bez nulq, napivreßitka idempo-
tentiv qko] ma[ skinçennu dovΩynu.
Todi S [ perestavnog v tomu i lyße v tomu vypadku, koly S — hrupa.
Dovedennq. Lehko perekonatysq, wo mnoΩyna K = { x ∈ S rank( )x = 0} [
hrupog, qka bude qdrom napivhrupy S. Oskil\ky za umovog napivhrupa S ne
mistyt\ nulq, to hrupa K ne [ tryvial\nog, tobto | K | ≥ 2. Na pidstavi lemy 8
(dyv. [2]) i teoremy53 (dyv. [9]) bezposeredn\o robymo vysnovok, wo K =5S, tobto
S — hrupa.
4. Xarakterystyka napivreßitky idempotentiv perestavno] inversno]
napivhrupy skinçennoho ranhu z nulem. Cilkom pryrodno posta[ pytannq: qki
neobxidni i dostatni umovy treba naklasty na napivreßitku L, wob vona bula
napivreßitkog idempotentiv deqko] perestavno] inversno] napivhrupy skinçenno-
ho ranhu.
Spoçatku rozhlqnemo najprostißyj vypadok, a same, koly napivreßitka L
ma[ dovΩynu, wo ne perevywu[ 1.
Teorema 4. Bud\-qka napivreßitka L , dovΩyna qko] ne perevywu[ 1, [ na-
pivreßitkog idempotentiv deqko] perestavno] inversno] napivhrupy z nulem.
Dovedennq. Rozhlqnemo antyhrupu Φ ( L ) — inversnu napivhrupu vsix izomor-
fizmiv miΩ holovnymy idealamy napivreßitky L . Lehko dovesty, wo antyhrupa
Φ ( L ) [ napivhrupog Brandta, qka, qk vidomo, [ perestavnog. Krim toho, oçevyd-
no, wo napivreßitka idempotentiv antyhrupy Φ ( L ) izomorfna napivreßitci L.
Teper rozhlqnemo vypadok, koly dovΩyna napivreßitky bil\ßa abo do-
rivng[52.
Teorema 5. Nexaj napivreßitka P ma[ skinçennu dovΩynu, qka ne menßa
niΩ 2. Napivreßitka P [ napivreßitkog idempotentiv deqko] perestavno]
inversno] napivhrupy z nulem todi i til\ky todi, koly vykonugt\sq taki dvi
umovy:
1) qkwo rank( )a = rank( )b , to aP ≅ bP;
2) dlq bud\-qkoho z ∈ P ( rank( )z ≥ 2 ) isnugt\ x i y taki, wo x < z, y <
< z, x ≠ y i rank( )x = rank( )y = rank( )z – 1.
Dovedennq. Nexaj vykonugt\sq umovy 1 i 2. Rozhlqnemo antyhrupu Φ ( P )
— inversnu napivhrupu vsix izomorfizmiv miΩ holovnymy idealamy napivreßitky
P. Oçevydno, wo napivreßitka idempotentiv antyhrupy Φ ( P ) izomorfna P.
Krim toho, Φ ( P ) mistyt\ nul\. Umova 1 zhidno z teoremog51 (dyv. [15]) zabezpe-
çu[ linijnu vporqdkovanist\ idealiv antyhrupy Φ ( P ), a otΩe, dlq napivhrupy
Φ ( P ) vykonu[t\sq umova 1 teoremy52. Takym çynom, za teoremog52 antyhrupa
Φ ( P ) [ perestavnog.
Navpaky, qkwo my ma[mo perestavnu inversnu napivhrupu S z nulem, napivre-
ßitka idempotentiv qko] ma[ skinçennu dovΩynu, pryçomu S mistyt\ element,
ranh qkoho ne menßyj niΩ 2, to za teoremog52 napivreßitka idempotentiv napiv-
hrupy S zadovol\nq[ umovu 2. PokaΩemo, wo vykonu[t\sq j umova 1. Dijsno,
nexaj a i b — idempotenty napivhrupy S, pryçomu rank( )a = rank( )b . Todi za
teoremog52 SaS = SbS, a otΩe, za lemog53 aE ≅ bE (E — napivreßitka idempo-
tentiv napivhrupy S).
5. Pryklady i ilgstraci]. Teorema52 da[ zruçnyj kryterij perevirky in-
versno] napivhrupy skinçennoho ranhu z nulem na naqvnist\ u ne] vlastyvosti ko-
mutatyvnosti konhruencij. Pered tym qk navesty konkretni pryklady we raz
nahada[mo, wo umova 1 (dyv. formulgvannq teoremy52) ekvivalentna linijnij
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1360 V. D. DEREÇ
vporqdkovanosti (vidnosno vklgçennq) idealiv napivhrupy.
Pryklad 1. Nexaj V — skinçennovymirnyj linijnyj prostir. Poznaçymo
çerez Aut p V( ) inversnu napivhrupu vsix çastkovyx avtomorfizmiv vektornoho
prostoru V. Oçevydno, wo Aut p V( ) [ napivhrupog skinçennoho ranhu i mistyt\
nul\. Vidomo, wo idealy napivhrupy Aut p V( ) linijno vporqdkovani, tobto vy-
konu[t\sq umova 1. Oçevydno, wo napivreßitka idempotentiv inversno] napiv-
hrupy Aut p V( ) izomorfna napivreßitci pidprostoriv linijnoho prostoru V .
Pislq c\oho zauvaΩennq lehko pereviryty, wo dlq napivhrupy Aut p V( ) vyko-
nu[t\sq j umova 2. OtΩe, Aut p V( ) [ perestavnog inversnog napivhrupog.
Pryklad 2. Nexaj N = {1, 2, 3,5…5,5 n5}. Poznaçymo çerez IO ( N ) inversnu
napivhrupu vsix vza[mno odnoznaçnyx peretvoren\, wo zberihagt\ porqdok (v da-
nomu vypadku zvyçajnyj porqdok na mnoΩyni N), a çerez IS ( N ) symetryçnu in-
versnu napivhrupu na mnoΩyni N. 5Oçevydno, wo IO ( N ) ⊆ I S ( N ) . Bud\-qka
inversna napivhrupa A taka, wo IO ( N ) ⊆ A ⊆ IS ( N ) , [ perestavnog.
Proilgstru[mo teoremu55 za dopomohog diahram.
Rys. 1
Na rys. 1 zobraΩeno diahramu napivreßitky, qka ne [ napivreßitkog Ωodno]
perestavno] inversno] napivhrupy z nulem. Tut ne vykonu[t\sq ani umova 1, ani
umova 2 (dyv. teoremu55).
Rys. 2
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
XARAKTERYSTYKA NAPIVREÍITKY IDEMPOTENTIV PERESTAVNO} INVERSNO} .… 1361
Na rys. 2 navedeno diahramu napivreßitky, dlq qko], oçevydno, vykonugt\sq
umovy 1 i 2 (dyv. teoremu55). OtΩe, cq napivreßitka [ napivreßitkog idempoten-
tiv perestavno] inversno] napivhrupy z nulem. V qkosti tako] napivhrupy moΩna
vzqty napivhrupu vsix izomorfizmiv miΩ holovnymy idealamy dano] napivreßitky.
Rys. 3
Na rys. 3 zobraΩeno diahramu napivreßitky, dlq qko] vykonu[t\sq umova 2,
ale, oçevydno, ne vykonu[t\sq umova 1. OtΩe, cq napivreßitka ne moΩe buty na-
pivreßitkog perestavnog inversno] napivhrupy z nulem.
Rys. 4
Na rys. 4 zobraΩeno diahramu napivreßitky, dlq qko], oçevydno,
vykonugt\sq umovy 1 i 2 teoremy55. OtΩe, cq napivreßitka [ napivreßitkog
idempotentiv deqko] perestavno] inversno] napivhrupy z nulem. Oçevydno, wo
dana napivreßitka ne zadovol\nq[ umovu Ûordana – H\ol\dera.
1. Klyford A., Preston H. Alhebrayçeskaq teoryq poluhrupp: V 2 t. – M.: Myr, 1972. – T. 1, 2.
2. Tamura T. Commutative semigroups whose lattice of congruences is a chain // Bull. Soc. Math.
France. – 1969. – 97. – P. 369 – 380.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1362 V. D. DEREÇ
3. Schein B. M. Commutative semigroups where congruences form a chain // Bull. Acad. pol. sci. Sér.
sci. math., astron. et phys. – 1969. – 17. – P. 523 – 527.
4. Schein B. M. Corrigenda to „Commutative semigroups where congruences form a chain” // Ibid. –
1975. – 12. – P. 1247.
5. Mitsch H. Semigroups and their lattice of congruences. I // Semigroup Forum. – 1983. – 26. – P. 1
– 63.
6. Mitsch H. Semigroups and their lattice of congruences. II // Ibid. – 1997. – 54. – P. 1 – 42.
7. Nagy A., Jones Peter R. Permutative semigroups whose congruences form a chain // Ibid. – 2004. –
69, # 3. – P. 446 – 456.
8. Trotter P. G., Tamura T. Completely semisimple inverse ∆-semigroups admitting principal series
// Pasif. J. Math. – 1977. – 68, # 2. – P. 515 – 525.
9. Hamilton H. Permutability of congruences on commutative semigroups // Semigroup Forum. –
1975. – 10. – P. 55 – 66.
10. Cherubini A., Varisco A. Permutable duo semigroups // Ibid. – 1984. – 28. – P. 155 – 172.
11. Bonzini C., Cherubini A. Medial permutable semigroups // Coll. Math. Soc. Janos Bolyai. – 1981.
– 39. – P. 21 – 39.
12. Jiang Z. LC-commutative permutable semigroups // Semigroup Forum. – 1995. – 52. – P. 191 –
196.
13. Jiang Z., Chen L. On RDGCn-commutative permutable semigroups // Period. math. hung. – 2004.
– 49, # 2. – P. 91 – 98.
14. Bonzini C., Cherubini A. Permutable regular ω-semigroups // Boll. Unione mat. ital. – 1988. – 7. –
P. 719 – 728.
15. Dereç V. D. Pro perestavni konhruenci] na antyhrupax skinçennoho ranhu // Ukr. mat. Ωurn.
– 2004. – 56, # 3. – S. 346 – 351.
16. Dereç V. D. Konhruenci] perestavno] inversno] napivhrupy skinçennoho ranhu // Tam Ωe. –
2005. – 57, # 4. – S. 469 – 473.
17. Derech V. On permutable inverse semigroups of finite rank // 5th Int. Algebraic Conf. in Ukraine:
Abstrs (Odessa, July 20 – 27, 2005). – Odessa, 2005. – P. 57.
OderΩano 06.09.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
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| id | umjimathkievua-article-3394 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:43Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e4/4cc636046217f89dbbee5d7f3056c6e4.pdf |
| spelling | umjimathkievua-article-33942020-03-18T19:53:10Z Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero Характеристика напіврешітки ідемпотентів переставної інверсної напівгрупи скінченного рангу з нулем Derech, V. D. Дереч, В. Д. We give a characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero. Дана характеристика полурешетки идемпотентов перестановочной инверсной полугруппы конечного ранга с нулем. Institute of Mathematics, NAS of Ukraine 2007-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3394 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 10 (2007); 1353–1362 Український математичний журнал; Том 59 № 10 (2007); 1353–1362 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3394/3531 https://umj.imath.kiev.ua/index.php/umj/article/view/3394/3532 Copyright (c) 2007 Derech V. D. |
| spellingShingle | Derech, V. D. Дереч, В. Д. Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero |
| title | Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero |
| title_alt | Характеристика напіврешітки ідемпотентів переставної інверсної напівгрупи скінченного рангу з нулем |
| title_full | Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero |
| title_fullStr | Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero |
| title_full_unstemmed | Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero |
| title_short | Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero |
| title_sort | characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3394 |
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