Retractive MV-algebras

Math w are&SoftC om puting 2( 19 95)157-165 Retractive MV -Algebr as 3 Ro bertoCignoli a &A n toniT orre ns b a De pto.Mate m  atica s.Fac. Cie ncia sEx actasyNa t. Univ.BuenosAires .Ci ud ad Univ ersitaria. 1428Bue nosAi re s(Argen ti na ) e-mail:p o stma st @cign ol.ub a. ar b Fac. Mate m  atique s.Univ .de Barc el ona . Gr an Vi a585.0800 7Ba rc elo na(Spain) e-ma il:to rr ens@c erb er.m at.ub.es Gi v e nalg e bras A and B ofthesam et ype ,a hom om orphi sm  : A ! B isc al led r etr active pro v idedthatthereisahom om orphi sm  : B ! A suc hthat    = id B .No tethataretracti v ehom omo rphi sm m ustbe surjec tiv e.Ac ongruencerelation  on A i scall ed r etr acti ve whent heca nonica lproje ction   : A ! A = i sr etra ctiv e.Analgebra A i s r etr a ctive pro vi de dallitscongruenc erelati on sarere trac tiv e. Ingrou ps,congrue ncere lationsc an be ide n tie dwi thnorm alsubgroups.An or ma ls ubgroup N ofagroup G is retracti v ei fandonlyi f G splitsover N (see [8 , x 15.1]).Thenitfoll ow st hat N i sre trac tiv ei f ando nlyifi tiscom ple m en t edinthelattic eofallsubgroupsof G .A n a nalogo us re su l t h olds fo r Bo o l ean algebras : Anide al I of a Bo ole an Al geb r a B is r etr act i ve i fan d onl y if the sub alg ebr a gen er ate dby I is c omple m e nte d in the l attic eof al l sub alg eb r as of B (see [1]). Thi s pap er is a rst app roac h to the theory o f retractiv eMV - algebra s. 3 Thisw ork ha sbeenm ade d uri ng the sta yo f the rst author in the \Ce n tre de Rec erc a Matematica de l'Institut d'Es tudis Catalans " and the Depart amentde Logica,Hist oriaiFil. de laCiencia of the Unive rsityofBarcelona (Spain). The s econd author ispartially supp ort ed by GrantPB94-0920 ofD.G.I.C.Y.T. of Spain. 157 158 R.Cignoli&A .T orrens We as sum efam il iarit ywi th thet heoryofM V-al ge bras,as de ve lope di n[ 3,4,9] (s ee also[ 10,6,5]).F orther eaderc on v eni ence, i nSec tion 1w ei ncludeafewresultson MV-a lg e brasthatareused i nt herem ai nde rofthepap er,a nd w ec onside rtherel ati on be tw e en retr acti vi ty andc om pl em e n tationint helatticeo fsubalgebrasofan MV -a lg eb r a.W eobtainthatanideal I of theMV-algebra A isretracti v ei fandonl yifthes ub algebraof A gene rate db y I isc om ple m en ted i nthelattic eo fs ubalgebrasof A .I nSe ction 2, we i n v e stigatetheretractivit yofdi rec tproductsof MV-algebrasandw ec harac terizet he nitere trac tiv eMV -algebras. Theauthorswi sht oe xpressthei rgrati tudetot heRefe reeand toS.Sessafor im pro vi ngtheori gi nalv ersion of Theorem1.2andb y suggesti ngtheproo fp re se n tedhere . 1MV-algebras .Retrac tive ide al s. A n MV- al gebra i sanalgebra A =( A; 8 ; : ; 0)oft y pe(2 ; 1 ; 0)satisfyi ngthefoll o wi nge quati ons: MV 1. ( x 8 y ) 8 z  x 8 ( y 8 z ) MV2. x 8 y  y 8 x MV3. x 8 0  x MV4. : ( : x )  x MV5 . x 8: 0 : 0 MV 6. : ( : x 8 y ) 8 y : ( x 8: y ) 8 x Bytaki ng y = : 0i nM V6, w e deduce : MV 7. x 8: x : 0. Theref ore ,if w es et 1 = : 0and x  y = : ( : x 8: y ), then ( A; 8 ;  ; : ; 0 ; 1) s ati se s all the a x iom sgi veni n[9 ,Le mma2. 6. ], and hence the ab ove denitio n of MV-algeb ras is equivalent to Cha ng's denition [3] (seealso [5, 6]). Retracti v eM V-algebra s 159 In thelanguageofMV -algebrasw ec onsi derthefol lo wi ng term s: x _ y = def ( x : y ) 8 y;x ^ y = def ( x 8: y )  y: F oreac hM V-algebra A ,there duc t L ( A )=( A ; ^ ; _ ; 0 ; 1) is abounde d distributiv el atti ce,withl easte lement 0andgreatestele m ent 1. Th e correspondingorderre lation,whic hw ecallthe naturalo rder of A ,i s gi v enb y x  y if an do nl yif : x 8 y =1(ore quiv alen tly , x : y =0). An MV -a lgebrasuc hthati tsnatura lo rderi stotaliscalle dan MV - c hai n . Le t A beanMV -al ge bra.Asubset I of A isc al led ideal pro vi de d that: (I 1) 0 2 I , (I 2) a 2 I an d b 2 I im pl y a 8 b 2 I , (I 3) a  b and b 2 I im ply b 2 I . Le t I ( A )and Con ( A )denote ,respe ctiv el y ,theseto fi deal sof A andthesetofc ongruencerelationson A .Thecorrespondenc e:  7! J (  )=0 = = f a 2 A jh a; 0 i2  g establ ishesa norder-isom orphism J from Co n ( A )on to I ( A ),wi thbo th setsordere dby incl usi on.Thein v er seof J is givenb y: J 0 1 ( I )= fh a;b i2 A 2 :( x : y ) 8 ( y : x ) 2 I g ; (see[ 3, 5,6]).F oreach idea l I of A ,we wri te A / I inplace of A / J 0 1 ( I ), and w e denote the eq ui v alenc e class of a n el em ent a 2 A ,b y a=I . An ideal I of an M V-algebra A is call ed retracti ve if the a sso ci - ate dc on gruence re lation J 0 1 ( I )i sre tra c tiv e. Le t A be an M V-a l ge bra . The uni vers e s of the suba l ge bra s of A , o rde red b yi ncl usi o n, form a lattice with sm all est ele men t f 0 ; 1 g and gre ate st ele men t A , tha t w e denote b y Sub ( A ). G i ven a n i deal I of A ,w e repre se ntb y h I i the univ erse of the s ubalgebra gene ra te db y I , i.e., h I i = I [: I , where : I = f: x j x 2 I g .In o rder to state the main result of th issection we recall a resultgiven in [7]. 160 R.Cignoli&A .T orrens Le mm a1.1 [7,L emma1 .6]L et A b ea nM V-a lgebr aand I anideal of A .Thenforany a;b 2 A thefollowingar ee quiva le nt : i ) h a;b i2 J 0 1 ( I ) ii) a =( b  h ) : k ,forsome h;k 2 I . 2 Rem ark: T osho wthati ntheab ov el e m m ai)im plie si i)itsuce st o tak e k = : a  b and h = a : b (see[7] ). Theore m1.2 An ideal I inanM V-algebr a A isre tr activeifandonly if h I i isc omplem ente dinthelattice Sub ( A ) . Proof: Let I beare trac tiv ei de alof A ,a ndle t  I bethee mb edding assoc iatedtothere trac tive projec tion  I .Cl early  I ( A=I ) \h I i = f 0 ; 1 g . L e t a 2 A ,since h a ; I ( a= I ) i2 J 0 1 ( I ),b yL em m a1. 1, a = (  I ( a=I ) 8 h ) : k forsom e h;k 2 I .He nce a 2  I ( A=I ) W h I i .Th us A =  I ( A =I ) W h I i and  I ( A =I )isthecom ple m en to f h I i i n Sub ( A ). Con v e rse ly ,assum ethat I isanidealin A suc hthat h I i isc om plem e n tedin Sub ( A ).Let S beacom ple m en to f h I i .Then (1) S \h I i = f 0 ; 1 g ,and(2) S W h I i = A . Co ndi tion(1)i m pl iesthatther estri ction  I  S is aone -to-onehom om orphismf rom S to A =I .Byconditi on(2), fore ac h a 2 A ,thereis aterminthela nguageofMV -al ge bras,sa y p ( x 1 ;:::;x m ;y 1 ;:::;y n ), suc hthat a = p A ( a 1 ;:::;a m ;b 1 ;:::;b n ) forsom e a 1 ;:::;a m 2 S andsom e b 1 ;:::;b n 2h I i .The n a =I = p A = I ( a 1 =I;:::;a m =I;b 1 =I;:::;b n =I ) : Sinc e b i =I =0 =I or1 =I ,w eha v ethat the reisn-tupl e ( t 1 ;:::;t n )of 0's a nd 1's suc h that a=I = p A ( a 1 ;:::;a m ;t 1 ;:::;t n )/ I , and si nce ( a 1 ;:::;a m ;t 1 ;:::;t n ) 2 S n + m , we ha ve that p A ( a 1 ;:::;a m ;t 1 ;:::;t n ) 2 S .There fo re the restric tion  I  S isan isom orphism from S on to A =I , a nd w e can ta ke  =(  I  S ) 0 1 . 2 For an yMV - algebra A , B ( A ) denotes t he B o olean algebra of all com pl em ented el em e nts i n L ( A ). Si nce for an y a 2 A and b 2 B ( A ), a 8 b = a _ b and a  b = a ^ b , B ( A )is a s ubalgebra o f A (s ee [3, 10, 6 ]) in which : b is the complementof b .Thenwehave: Retracti v eM V-algebra s 161 Theore m1. 3 I f A isar etr activ eM V-algebr a,then B ( A ) isr et ract ive Bo ole analgebr a. Proof: Let I beanont ri vi ali dealof B ( A ).B y1.2 ,t op ro v ethat I i s retractiv ei tsuc estosho wthat h I i i sc om ple m en tedi n Sub ( B ( A )). Denoteb y( I ]thei dealg enerat edb y I i n A andl et  : A ! A = ( I ]the naturalprojec tion. Si nce A i sre trac tiv e,therei sa m onom orphi sm  : A = ( I ] ! A suc hthat    = id A= ( I ] .Itise asyt oc hec kthat a 2 B ( A )im pl ie s  ( a = ( I ]) 2 B (  ( A = ( I ] )),hence B (  ( A = ( I ] ))  B ( A ). Therefore I \ B (  ( A = ( I ]))  ( I ] \  ( A = ( I ])= f 0 ; 1 g : Ontheot he rh an d,foran y a 2 I , h a; ( a= ( I ]) i2 J 0 1 (( I ] ). B yt he rem arkfol lo wingL em m a1.1,the reare k;h 2 ( I ] \ B ( A )= I suc h that a =(  ( a= ( I ] ) 8 h ) : k .H ence B ( A )= I _ S ub ( B ( A ) ) B (  ( A = ( I ])) : Th us B (  ( A = ( I ] ))i sthec om plem en tof h I i i n Sub ( B ( A )). 2 2Produ ctsof retractiv eMV-algeb ras W esa ythatt he algebras A and B are c om pa tible iftherearehom om orphism s ' : A ! B and : B ! A . Int he nex tl em m aw ecol le ctsom ei mm edi atec onseque ncesthe denitio no fa retracti v ealgebra. Le mma2. 1 The f ol l owing pr op erties hold for e ac h r etr activ e alg ebr a A : (i) Each n on trivial hom omorph ic image of A is a r etr ac tive alg ebr a . (i i) Any two n on triv ial homomor phic images of A ar ecom patibl e. (ii i) If A is a sub dir ect pr oduct of a famil y f A i g i 2 I of non t r ivial a lg ebras, then al l the alg eb ras A i areret ractiv e an d pairwise compat ible . 2 162 R.Cignoli&A .T orrens Le t[ 0 ; 1 ]= h [0 ; 1] ; 8 ; : ; 0 i ,wh ere[ 0 ; 1]de notestheuni ts eg ment of therealli ne,a nd theo perations 8 and : aredened by thepre sc ripti ons a 8 b =mi n f 1 ;a + b g and : a =1 0 a .I ti sw el lk no wnthat [0,1] and i tssubalgebrasaretheo nl ys im pl eMV -al gebras.Th en it fo l lo ws that thei dentit yi st heonlyend omorph ismo fasimpleMV-a lgebr a . Lem ma 2.2 Thefollowingpr op erties holdfore achr etr activeMV - algebr a A : (i ) If J; K a r emaxi malide also f A ,then A =J  = A =K . (ii ) If A i sasub dir e ctpr o duc tofafamily f A i g i 2 I ofsimpleM V- algebr a s,then fora ll i ;j 2 I , A i  = A j . Proof: (i)Si nce A =J and A =K arehom om orphici m ages of A ,b y i tem(i i)inLem m a2.1,they ar ec om pati ble.N o wthere sul tf ol lo ws fromt hefactthattheiden t it yi sth eonl yendom orphi smofasi m pl e MV -al ge bra . (i i)follo wsfrom( iii )inLe m m a2. 1and(i ). 2 Lem m a2.3 If A 1 ;:: :; A n are MV-algebr as,th ene achide al J o f A = A 1 21112 A n i softhef or m J = J 1 21112 J n ,wher e,f or e ach i =1 ;:::;n , J i isanide alof A i . Proof: Let J beanidealo f A ,andforeac h i =1 ;:::;n ,let J i =  i ( J ), where  i de note stheproje cti ono nto A i .Itisplainthat J i isani deal of A i and that J  J 1 21112 J n .Supp osethat a =( a 1 ;:: :;a n ) 2 J 1 21112 J n .Thi si m pli esth at b 1 =( a 1 ; 0 ;:::; 0), b 2 =(0 ;a 2 ; 0 ;:::; 0), ::: , b n =(0 ;:::; 0 ;a n )are i n J ,andthen a = b 1 8111 8 b n 2 J . Theref ore J = J 1 2111 2 J n . 2 Theore m2.4 If A 1 ;:::; A n den ote non trivial MV-al geb r as, the n A = A 1 2 111 2 A n is r etr act ive if an d onl y if al l the A i ar er etr act ive and pairwise c omp atibl e. Pro of: The onl y if part is a pa rti cular cas e of ite m(ii i) in L e mma2. 1. Toprovethe if part, supp ose that for i =1 ;:::;n , the na tural pro - jection % i : A i ! A i =J i hasarightinverse  j : A i =J i ! A i ,andthat Retracti v eM V-algebra s 163 fore ac h1  i;j  n ,therei sahomomorphi sm ' ij : A i ! A j .L et J beaprope ridealo f A .B yLe m m a2. 3, J = J 1 21112 J n ,where, foreach i =1 ;:::;n , J i i sanide al of A i .Since J is ap rope r ide alof A ,theset T = f i 2f 1 ;::: ;n gj J i 6 = A i g is nonem pt y ,sa y T = f i 1 ;:::;i k g .Then A =J  = A i 1 =J i 1 21112 A i k =J i k ,andw ecan iden tify thenaturalprojection % J wi th them appi ng( a 1 ;:::;a n ) 7! ( % i 1 ( a i 1 ) ;:::;% i k ( a i k )).Thehom om orphi sm  : A i 1 =J i 1 21112 A i k =J i k ! A gi v e nb y  ( a 1 =J i 1 ;:::a k =J i k )=( b 1 ;:::;b n )where,foreac h j = 1 ;: ::;n , b j isg iv en b ytheprescripti on: b j = (  i r ( a r =J i r )if j = i r 2 T ' i 1 j ( a 1 ) if j 62 T isari gh ti n v erseof % J .There fore , J isare tra ctiv ei dealof A . 2 Corol lary 2.5 F ore achr etr activeM V-alge br a A ande achi nte ger n  1 , A n isar etr activeM V- algebr a. 2 W ede noteb y L n + 1 thesubalgebraof [0, 1] ,whoseuniv erseist he subset L n +1 of[ 0 ; 1]form edb ythe r ati onalf rac tionswi thdenom inator n +1.Theseal gebrasarethe onl yni teMV -c hains. Since e ac hsi m pl eal ge braisretracti v e ,theabo v ecorol laryi mpli es that e a chnitep ower ofas impleMV-a lgebr aisre tr active .H ence ,b y tak ingi n toacc oun tthat ea ch niteMV -al ge br aisad irectp roductof sim pl eMV-algebras(see[10,6 ]) an ditem(i i)inLem m a2.2,w eo btai n thefollo wi ngc haracteri zationofr etracti v eni teMV -algebras . Theore m2. 6 Anite MV-a lgeb ra A isr etr activeif ando nl yi ft he r e are natur alnumb ers n> 1 and r> 0 suchthat A  = L r n . 2 Rem ark: Si nce L 2 3 con tains a cop yof L 2 2 L 3 ,w esee tha t asub alg ebr a of a r etr ac tive MV- a lgebr am ay b enonr etr act ive . Our next task is to s ho w tha t t he restric tion to nit efam il iesi n Theo re m2.4 i si ndeed nece s sary . Theorem2.7 Thedirect produc t of an in nit e family of non trivial MV-al ge bras is not retrac tive . 164 R.Cignoli&A .T orrens Proo f: Let f A i g i 2 I be af am il yofnontriv ialMV-a lg eb ras.Sinc e B ( Q i 2 I A i )= Q i 2 I B ( A i ) ,itfoll o wsfrom Theorem1. 3thattopro v e thetheoremi ts u cestopro v ethefollowin g: Claim : The direc tproductofaninnitef am ilyofnontriv ialBo olean algebrasi snotre tra ctiv e. T opro v ethecl aim, let f B i g i 2 I beafam il yofnontrivia lB oolean algebras ,a nd B = Q i 2 I B i .I f I is an inni teset,then I con tainsa n in ni teden um erablesubset, sa y D = f i 1 ;:::;i n ;i n +1 ;: :: g .F ore ac h n  1,l et b n betheel em e n to f B denedb ythepresc ription: b n ( i )=  1i f i = i n 0 otherwise and b 0 betheelemen tdenedb y b 0 ( i )=  1i f i 2 I n D 0i f i 2 D F ore ac h n  0,l et U n bean ultralte rof B whic hcon tains b n , andle t ! bet hesetofnaturaln um bers.Thenthecorrespondenc e x 7!f n 2 ! j x 2 U n g de ne sahom om orphismfrom B on to P ( ! ), theBool eanalgebraofa ll subsetsof ! (seetheproofo f[ 2,Corol lary 2.17,p.4 07]).Sinc etheidealof P ( ! ) for m edb ythenitesubsets of ! isnotretra ctiv e(see[2,Proposi tion 2.16 ,p .407] ),w eha v ethat B hasanonre trac tive ho mo m orphi cim age,andthe n, by it em(i )in Lem m a2.1, B isnotretracti ve . 2 Coroll ary 2. 8 The dire ctpr o ductofafamil y f A i g i 2 I ofnontr iv ial MV-algebr asis r etr a ctiveifa ndonlyifthei ndexset I isni teandall the alg ebras A i ar er etr act ive and p air w ise c omp at ible. 2 Refere nces [1] R. B onne t, S ubalgebras, in [2] v ol 2, 3 89 -41 6 [2] R. Bo nnet a nd D. Monk, Handb ook of Boolean algebra s vols. 1, 2 and 3, Norht-Holland, 1 989 . Retracti v eM V-algebra s 165 [3]C . C.Chang,A lgebrai ca na lysisofm an y -v alue dl ogic s, Tran s. Am e r.Math.Soc . . 88 (1 958),467-490. [4]C. C.Chang,Ane wproo fo ft he c om plete nessofthe Luk as ie wicz axi om s, T ran s. Am er.M ath.Soc. 93 (1959), 74 -8 0. 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