NON-IDEMPOTENT PLONKA FUNCTIONS AND WEAKLY PLONKA SUMS

Transcription

1 DS Davdova YuM Movssa УДК 557 NON-IDEMPOTENT PLONKA UNCTIONS AND WEAKLY PLONKA SUMS DS Davdova YuM Movssa Russa-Armea Uvers Yereva Armea Yereva Sae Uvers Yereva Armea We roduce coceps o a o-dempoe Ploka uco ad a weakl Ploka sum ad esablsh a approprae correspodece bewee hese objecs I he las par o he paper we cosruc eamples o o-dempoe Ploka ucos ad he correspodg weakl Ploka sums I parcular we prove ha ever weakl dempoe quaslace s a weakl dempoe lace or a weakl Ploka sum o weakl dempoe laces Kewords: weakl dempoe semlace weakl dempoe lace hperde o-dempoe Ploka uco weakl Ploka sum weakl dempoe quaslace Ackowledgeme The work o secod auhor s suppored b he Sae Commee o Scece o he Republc o Armea gras: 0-3/-4 ad 5T-A5 00 Mahemacal Subjec Classcao: 03G0 06A 06B05 06B006B5 06B0 0A05 0B6 0C05 03C5 РУССКАЯ АННОТАЦИЯ В статье вводятся понятия неидемпотентной функции Плонка и слабой функции Плонка а так же устанавливается связь между этими объектами В последней части работы нами построены примеры неидемпотентных фукций Плонка и соответствующих им сумм Плонка В частности доказано что каждая слабо идемпотентная кваирешетка является либо слабо идемпотентной решеткой либо суммой Плонка слабо идемпотентных решеток КЛЮЧЕВЫЕ СЛОВА: слабо идемпотентная полурешетка слабо идемпотентная решетка сверхтождество неидемпотентная функция Плонка слабая сумма Плонка слабо идемпотентная квазирешетка Iroduco There es varous eesos o he cocep o lace or eample works [] [] weakl assocave laces were roduced I [3] a algebra wh a ssem o dees was roduced whch we call weakl dempoe laces see also [4] [5] I [6] he coceps o p- uco ad o drec sum o algebras wh lub-proper were roduced Ad here was proved ha here s a bjecve correspodece bewee ever p-uco o he algebra U ad s represeao as a drec sum o algebras wh lub-proper I hs paper we roduce coceps o a o-dempoe Ploka uco ad a weakl Ploka sum ad esablsh a approprae correspodece bewee hese objecs I he las par o he work we cosruc eamples o odempoe Ploka ucos ad he correspodg weakl Ploka sums I parcular we prove ha ever weakl dempoe quaslace s a weakl dempoe lace or a weakl Ploka sum o weakl dempoe laces Deo The algebra L wh oe bar operao s called weakl dempoe 7

2 Вестник КГЭУ semlace sases he ollowg dees: a b b a commuav a b c a b c assocav a b b a b weakl dempoec 3 The operao s called produc Addg he dempoe de a a a o we oba a semlace The eleme a L s called dempoe o he weakl dempoe semlace L a a a The se o he dempoe elemes o each weakl dempoe semlace orms a semlace e he produc o a wo dempoe elemes a weakl dempoe semlace s a dempoe eleme Deo The algebra L wh wo bar operaos s called weakl dempoe lace he reducs L ad L are weakl dempoe semlaces ad he ollowg dees are vald: a ba a a a b a a a weakl absorpo 4 a a a a equalao 5 or eample he ollowg algebra Z \ {0} where ad [ ] or whch ad [ ] are he greaes commo dvsor gcd ad he leas commo mulple lcm o ad s a weakl dempoe lace whch s o a lace sce ad or egave Le us recall ha a hperde s a secod-order ormula o he ollowg pe: X X m w w where X X m are ucoal varables ad are objec varables he words erms o w w Hperdees are usuall wre whou quaers: w w We sa ha he algebra Q he hperde w w s sased hs equal s vald whe ever objec varable ad ever ucoal varable s replaced b a eleme rom Q ad b a operao o he correspodg ar rom supposg he possbl o such replaceme [7]- [5] O characerao o hperdees o varees o laces modular laces dsrbuve laces Boolea ad De Morga algebras see [4] - [0] Abou hperdees hermal polomal algebras see [7]-[9] Noe ha ever weakl dempoe lace L sases he ollowg hperde erlaced hperde: X Y X Y Y X Deo 3 A bar algebra A U s called a weakl dempoe quaslace

3 DS Davdova YuM Movssa sases he ollowg hperdees: X Y 6 X X 7 X X X X X X X 9 X Y X Y Y X 0 Noe ha ever weakl dempoe lace sases he hperdees o hs deo No-dempoe Ploka ucos weakl Ploka sums Deo 4 A algebra A U s called a weakl Ploka sum o s subalgebras U where I he ollowg codos are vald c [5] [6] [0] []: U or all j I j U j I U U O he se o he dces I here ess a relao " " such ha semlace wh he ollowg properes v I j j k k j he here ess a homomorphsm : U U j j or j k ad or a v or all 9 I s a upper where ad U A ad or all Q he ollowg equal s vald: A A 0 0 where he ar A U U I sup{ } 0 I a algebra A U s a weakl Ploka sum o s subalgebras U where I he we wre A Sum U The de hperde s sad o be regular he se o objec varables occurrg s equal o he se o objec varables occurrg Theorem 5 Le A U be a weakl Ploka sum o s subalgebras U where I ad I The he algebra Sum U are sased all regular dees hperdees ha are sased each algebra U ad moreover ever oher de hperde s alse Sum U Proo The rs par o heorem obvousl ollows rom he deo Le us prove he secod par Le be a o regular de sased Sum U ad hece s sased each subalgebra U Le us deoe he se o varables occurrg b Var ad ha o

4 Вестник КГЭУ here s Var So here ess a varable j Var \ Var Sce I b I such ha Le us choose a j b j he equao A A we have: a A ad he b A ad subsue The obvousl A A ad sce Deo 6 Le A U be a algebra The bar operao : U U U s called a o-dempoe Ploka uco o U sases he ollowg dees c [6] [] []: or a operao 3 4 or a operao operao 5 or a operao 6 or a ad or a or a operao 7 To oba a o-dempoe Ploka uco dere rom Ploka uco oe s o assume ha o operao o he algebra A s dempoe Theorem 7 Le A U be a algebra wh a o-dempoe Ploka uco The A s a weakl Ploka sum o s subalgebras Proo Dee o he se U he relao U U he ollowg wa: a b a b a a b a b b where s a o-dempoe Ploka uco or A Le us show ha s a equvalece o U Ideed relev ad smmerc a b ad b c he a b a a b a b b b c b b c b c c mmedael ollow rom he deo Show rasv: le Hece: a c a a c a a c a b c a b c a b b 3 a b a a a b a b a a c a c c a c c a c b a c b a c b b 0

5 DS Davdova YuM Movssa Thus c b b c c b c c b c b c c a c Deog he correspodg equvalece classes b paro o U : { U I} U Le us prove ha subses U are subalgebras Ideed le a he ar we ge: U a a 6 7 I we oba a he or U I a a a a a a a a a 5 a a a a a a a a 7 a a a a a a a a a a a a a U e Noe ha or ever a bu we have: a b a b a b Ideed: 3 a b a b a b b a a b b a a b a a b Le us also oe ha rom he de a b a b a b b a ollows ha a b b a urhermore a a ad a b a b b b he Ideed: 3 3 a b b a a b a b a b b a a b b a 3 a b a a a b a a b a b a b a b I he same wa we ge ha: a b a b a b a b Moreover rom de o deo 5 mmedael ollows ha a a a or a a U O he se o dces I dee a order " " b he ollowg rule: here es such a U b U ha b a b b Ths order makes he se I a srucure o a semlace Ideed relev mmedael ollows rom he deo Le us show ha " " s asmmerc: Le ad he here es a a U b b U such ha

6 Вестник КГЭУ b a b b ad a b a a Hece: 3 a a b b a a b b a a b b a a b b 3 3 a a b b a a a b a a b a a a a a a I he smlar wa we ge ha b b a a b b b b So a a b b hece cosequel b b U U ad 3 Le The here es b a b b d c d d So: b b b b hus a U U b c U d U 3 such ha d b d d b d d b d c b d c b d c c d c c d d c d d c d d d d d hece: d a d d a d d a d b a d b a d b b d b b d d b d d b d d d d d whch proves ha 3 Thus I s a ordered se To show ha I s a semlace le a U b U j ad a b U The: k a b a a b a a a b a a b a b a b a b some Hece or a a U b U Le us assume ha or some j j I here ess a upper boud k I such ha a b U or l I l ad k j l e here are a U c U such ha c a c c ad here are b U d U such ha d b d d Hece we have: j c a b c a b c c b c d b c d b c d d c d d c c d c c l l

7 DS Davdova YuM Movssa Thus a b c ad rom he assero a b a b ha s prove above we oba ha a b c whch meas k l ad k sup j Dee he mappgs : U U or he ollowg wa: where a U bu rs o all le us show ha or all a b U a a b a U bu Sce he here es c U d U such ha d c d d Thus we oba : 3 d d c d c d d c d d ad 3 c d d c c d Ths gves b d c d a b ad hece a b U The deo o mappgs s cosse e s depede rom he choce o he U a b a b eleme b Ideed le be arbrar elemes rom U ad a U The: 3 a b a b a b a b b a a b b a 3 a b b a a b b a a b b a a b a b a b a Thus we have: a b a b I s clear ha he mappgs 3 3 are homomorphsms ad all we prove ha or a -ar operao ad 0 or U U where sup { 0 } To make he proo easer le us deoe We have alread oced ha or a U ad b U j a bu sup j Ths mples ha U 0 we have: B or each Hece b 5

8 Вестник КГЭУ Sce b 6 or each we oba ha 7 Ths meas ha ad cosequece 0 U Le 0 U The whch shes he proo Eamples I Le Q Q be a semgroup wh he dees: ad The uco Q Q Q : deed b he equal s a o-dempoe Ploka uco o Q ad hece b Theorem 4 Q s a weakl Ploka sum o s subalgebras II Le Q be a semgroup wh he dees: ad The he uco Q Q Q : deed b he rule s a o-dempoe Ploka uco or Q Ideed: hece

9 DS Davdova YuM Movssa Thus b Theorem 4 Q s a weakl Ploka sum o s subalgebras III Le A U A B be a weakl dempoe quaslace wh wo operaos Usg he ollowg hperdees: X Y X Y X Y X Y X Y X 3 Y X Y X X Y X Y X Y 4 X X Y Y Y X Y Y 5 X Y Y X 6 Y Y X Y Y Y 7 X Y X Y Y Y X Y X Y Y Y 9 ha are he cosequeces o he hperdees Ошибка! Источник ссылки не найден we show ha he uco : U U U deed b he rule A B s a odempoe Ploka uco o A Ideed: A B A A B B A B 77 A A B A B A B B B A A B A B A B B B A B 7 A A B B B A B A A B B B 4 5 A A B B B A B B A B A B A Y A B B 6 9 A B A A A 3 A B A B A B A B A B A B rom he hperde ollows ha urher whou loss o geeral we suppose ha A A A A B A A A A B B A A B 6 A A A Y A Y A A B A B 3 A A A B B A A B A A B 5 A A B A B A A

10 Вестник КГЭУ A A A B A B A A A B A A A A A A 7 A A A B B A A A A A A X A A A A 6 A B A B A B 9 6 A B B A Y A B A 6 B A A B 9 B A A Now usg Theorem 4 we ge he ollowg resul Theorem Ever weakl dempoe quaslace wh wo bar operaos s a weakl dempoe lace or a weakl Ploka sum o weakl dempoe laces Reereces [] E red Weakl Assocave Laces wh Cogruece Eeso Proper Algebra Uversals [] E red G Graer A Noassocaave Eeso o he Class o Dsrbuve Laces Pasc Joral o Mahemacs [3] II Melk Nlpoe sh o maolds Mah Noes [4] E Gracska O ormal ad regular dees Algebra Uversals [5] J Ploka O varees o algebras deed b dees o some spesal orms Houso Joural o Mahemacs [6] J Ploka O a mehod o cosruco o absrac algebras ud Mah [] YuM Movssa Blaces ad hperdees Proceedgs o he Seklov Isue o Mahemacs [] YuM Movssa Ierlaced modular dsrbuve ad Boolea blaces Armea Joural o Mahemacs [3] YuM Movssa Iroduco o he heor o algebras wh hperdees Yereva Sae Uvers Press Yereva 96 Russa [4] YuM Movssa Hperdees algebras ad varees Uspekh Ma Nauk Russa Eglsh rasl Russ Mah Surves [5] YuM Movssa Hperdees ad hpervarees Sceae Mahemacae Japocae [6] YuM Movssa Hperdees o Boolea algebras IvRossAcadNauk Ser Ma Russa Eglsh rasl RussAcad Sc Iv Mah [7] YuM Movssa Algebras wh he hperdees o he vare o Boolea algebras IvRossAcadNauk Ser Ma Eglsh rasl RussAcad Sc Iv Mah [] YuM Movssa VA Aslaa Hperdees o De Morga algebras Logc Joural o IGPL do:0093/jgpal/jr053 [9] YuM Movssa VA Aslaa Algebras wh hperdees o he vare o De Morga algebras Joural o Coemporar Mahemacal Aalss [0] YuM Movssa VA Aslaa Subdrecl rreducble algebras wh hperdees o he vare o De Morga algebras Joural o Coemporar Mahemacal Aalss [7] K Deecke J Kopp M-sold varees o Algebras Advaces Mahemac 0 Sprger- 6

11 DS Davdova YuM Movssa Scece+Busess Meda New York 006 [] K Deecke SL Wsmah Hperdees ad Cloes Gordo ad Breach Scece Publshers 000 [9] R Padmaabha P Peer A hperbase or bar lace hperdees Joural o Auomaed Reasog [] J Ploka A Romaowska Semlace sums Uversal Algebra ad Quasgroup Theor Helderma VerlagBerl [] A Romaowska JDH Smh Modes World Scec 00 Сведения об авторах Давидова Диана С Мовсисян Юрий М Auhors o he publcao DS Davdova Isue o Mahemacs ad Hgh Techolog Russa-ArmeaUvers 3 Hovsep Em sr Yereva 005 Armea YuM Movssa Deparme o Mahemacs ad Mechacs Yereva Sae Uvers Ale Maooga sr Yereva 005 Armea Дата поступления 506 7

12 Вестник КГЭУ D S Davdova Isue o Mahemacs ad Hgh Techolog Russa-ArmeaUvers 3 Hovsep Em sr Yereva 005 Armea E-mal:ddavdova@aderu YuM Movssa Deparme o Mahemacs ad Mechacs Yereva Sae Uvers Ale Maooga sr Yereva 005 Armea E-mal: movssa@suam