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SIMPLE OCKHAM ALGEBRAS

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SIMPLE OCKHAM ALGEBRASSIMPLE

    SIMPLE OCKHAM ALGEBRAS

plealgebra.Congruencelattic.2-elementchain

    ;1991MRSubjectClassification06B

    ;ChineseLibraryClassificationO1531

    ;AnOckhamalgebraisanalgebra(;AV,f.01)oftype(2,2,1,0,0)suchthat(L;A ;V,0,1)isaboundeddistributivelatticeandfisaunaryoperationdefinedonLSuchthat

    ;forall,?L,

    ;,(zA)=,(z)V,(),

    ;f(xV)=/(x)A,(),

    ;f(o)=1,f(1)=0

    ;ThusfisadualendomorphismonL.Thestudyofthesealgebrashasbeeninitiatedby

    ;J.Berman[jwhogaveparticularattentiontoceltainsubvarieties?whichisdefinedfor

    ;P?1,g?0bytheconditionf=,AnalgebraLiscalledsimple,ifConL=f,}.

    ;AsshownbyBerman[1thatifL?K

    ;PissimplethenLisfiniteandL?Kp,

    ;0Herewe

    ;shalldescribethestructureoffinitesimpleOckhamalgebras.

;ForanOckhamalgebra(;,)let

    ;c(L)={0,1}u{z?Llf(x)=z}

    ;For?,b?

    Linwhatfollowsweshallmeanthataandb&renotcomparablebynll6;

    ;otherwisebyafb.Wehavethefollowingresults ;Theorem1-LetLbeasimpleOckhanalgebra.a?Lissuchthata#,(0)then ;n?G1

    ;Proof.Assumefirstthatf2()aConsidertherelationddefinedby

    ;(z,)?告xAa=Aaandzv/(a)=V,(n).

    ;Let(,)?;then

    ;,()Afs(a)=f(y)A,(n).

    ;ManuscriptreceivedOctober23,1993 ;DepartmentofMathematics,ZhongshanUniversity,Guaugzhou510271,Ch

    ina

    ;

    ;214CHINANN.0FMATHV0L17SetB

    ;Sincef(a)?,itfollowsthat,(z)AB=f(v)Aa,andsowecanseethat.?

    ConLwith

    ;=’(n1)Alt(0,,(B))=lt(aA,(B),,(n))

    ;SinceLissimplebythehypothesis,welmveeither=/2orn=.Nowwhenn=

    ;wehave(0,1)?4andthen0Aa=1AB,whencea=0?e();wheno=uwehave ;aAf(a)=,(n),whencef(a)a.So

;f(a)Sa,.(B),,.(n),(B)?,.(B)

    ;andsoon,andwehavethesubalgebrachain ;0...?f2.+l(n)-..f(a)nf2(),‖(B)??1.

    ;SinceeverysubalgebraofasimpleOckhamalgebraisalsosimple,itfollowsfr

    omf3,Theorem

    ;4.3.21thatB?()

    ;Asimilarargumentholdsiff(a)a.

    ;Theorem2.LetL6e?simpleOckhamalgebra.a),thenA,(),aVf(a) ;G(1.

    ;Proof.Weshowfirsttlmtav/(~)e().Suppose,bywayofobtainingacontradicti

    on,

    ;thatavf(a)?e().Weconsiderthefollowingtwocases: ;(a)avf(a):1.

    ;ConsidertheprincipalcongTuencelat(0,B).For(z,)?l(0,B)wehave.v?

    =va.

    ;s0

    ;,(z)Af(a)=,()A,(B)and(,(.)A,(n))vn=(,()A,(n))VB.’

    ;SinceBv/(a)=1,itfollowsthat,)Va=,()VB.Consequently,(,(),,0))?1,l(0,

    ).

    ;Hence1,l(0,B)?ConL.SinceLissimpleanda?0,wehave(0,a)=andso ;(0,1)?ht(0,n).Itfollowsthat0va=1va,whencethecontradictionB=1.

    ;(b)Bvf(a)isafixedpoint.

;Inthiscasewehavef(a)Af2(B)=BV,(?).Sof(a)?andf2(),(n),and

    ;sof(a):,().SinceLissimple,fisinjective.Thenwemusthavef(a)=,other

    ;contradiction.

    ;Finally,bytheaboveobservationswehave,(.)vf2(B)e(),forf(a)e().Con- ;sequently,aA,(B)e().

    ;Theorem3.Let(f)?Kp.obefinite.Thenftakesatomsto~toms,and?nersely.

    ;Proof.LetabeanatomofLandletf(a)?<1.SinceLisfiniteandfinjectiveit

    ;followsthatfissurjeclive,thereexists=?Lsuchthat=f(z)andsof(a)f(z)<1.

    ;Sincefisadualautomorphism,wededucethata=>0.SinceBisanatom,itfollows

    ;that?=,whence

    =,(?),andsof(a)isacoatomDually,ifbisacoatomthenf(b1is ;anatom.

    ;Co~ollarly.Let;f)?Ko6esimple.n?Lisa?latomsoisf(n).

    ;WenowshallbeconcernedwithfiniteOckhanalgebrasThefollowingexamples,

    ;aswe

    ;shallsee,areoffundamentalimportance.

    ;Example1.LetLbethebooleanlattice2‖withatoms1,.

    ,an,InordertomakeL

    ;intoanOckhamalgebra,itsufficestodefinef(o)=1andtospecify,(Bt)foreachB;for

    ;eve’cyz?0CRnbeexpressedu~1quelyintheform2=VnwhereIisanon-emptysubset

    ;

    ;No2Fang,JSIMPLEOCKHAMALGEBRAS25Inparticular,wehave ;,.()=,(ni+)=,(V)=,)=+=nm.

    ;J?+1j?1j?i+1

    ;ItfoUowsthatifnisoddthenf.inducestheatomcycle ;n1_?n3-?n5_?…_?on_?o2_?…nn1_?o1,

    ;whereasifeventhenf.inducesthetwoatomcycles ;ol_?o3_?o5_??--n1_?o1;n2n4.?o_?o2

    ;(a)isodd.

    ;Supp;f)

    ;issimple.Notethatinthiscasetherearenofixedpoints;forifawereafixedpoint

    andnt

    ;isallatomwithntQthenai+2=,.(ai)dsoalltheatomswouldbecontainedina,

    ;whichisnotpossible.

    ;(b)niseven.

    ;Inthiscaselet

    ;=alVa3V-Valand=.2Va4v--vnn.

    ;ThenaA:0andav=1.andso

    ;a::

    ;An=AS(az):,(Vn.):,(a)f?ft?,t?,

    ;Thusisafixedpoint,andsimilarlysois.Arguingasincase(a),andusingthefactthat ;inthiscasetherearetwoatomcyclesunder,.,weseethateither ;(0,n)=0,V)?.r(0,)=0,V%+t)?.4?i4Ei

    ;Ineithercasewededucethat(0,1)?,where=andagain(2;,)issimple. ;Example2.Let2betheverticalsumoftwocopiesof2.Lettheatomsbe ;.,a2,’..,nnandletthecoatomsofLbe61,62---k.ThenwecanmakeLintoanOckham

    ;algebrabydefiningS(o)=1,fO)=oand,withreductionmodulonwhereapprop

    riate

    ;f(ai)=6,,(6)=o+1.

    ;Weextendftoadualendomorphismbydefining ;,(Vn):人机,,(6{)=ViEf?it?t6i

    ;Observethat(22;f)hasasinglefixedpoint,namely ;b

    ;=

    ;0V

    ;l_

    ;a

    ;

    ;2l6CHIN.ANN0FMATH.V0Ll7SetB

    ;Supposethatd?Con(232;,)issuchthat?,andlet(z)?口withz<.We

;considerthefollowingthreec~ses:

    ;(1)Y?【0d.InthiscasethereexistsaI1atomaksuchthataksYandak. ;Consequently(0,)=(akA,aA)?毋andthen

    ;(0,B+I)=(0,f2(.k))?d.

    ;Sincetheatomsfromasinglecycleunderf,itfollowsthat(0,.)?

    dforeveryatomm.

    ;Wethereforehave

    ;,

    ;n

    ;(o,n)=(0,V.;)?d,i=l

    ;whence(0,1)?口andso=.

    ;(2)<Y.Thiscaseisthesa/neas(1),thereisanatomakzand>ak,whence ;again=

    ;(3),Y?【Of,11.Inthiscasethereexistsacoatombkwithbk>andbkY.Thus ;(bk,1)=(bkVz,bkVY)?d

    ;andtherefore(b+I,1)=(f2(),1)?

    .Sincethecoatomsformasinglecycleunderf.,

    ;wehave(6;,1)?口foreverycoatom64Hence

    ;n

    ;(n,1)=(6,1)?.i=1

    ;Itfollowsagainthat(0,1)?口andthen=

    ;Wethereforehavefromthoseobservationsabovethat(212;f)issimple.

    ;Definition.i}LsafiniteOckhamalgebralthenasubalgebraALwillbecalleda

    ;―subalgebraAcontains0‖theatoms.,L

    ;Theorem4.LetLbeBfiniteOckhamalgebra.IfLcontains(2;f)?.H

    ;subalgebrathenLssimple.

    ;Proof.ClearlyLhasuniquefixedpointQandif8isanyat0mofLthenwehavefr

    om

    ;Theorem3thatevery,.(n)isanatomande0,Vn)?口

    ;

    ;N02Fang,SIMPLEOCKHAMALGEBRAS2l7 ;giredlA?uandso,sinceAissimple,I–VIAThen(0,1)?dIAandso(0,1)? ;Hence=andLissimple.

    ;Example3.Considerthelattice ;1

    ;0

    ;madeintooJ1Ockhamalgebrabydefining ;t:01nbedefP1(PzY

    ;f(t1:10(PP6cafdeYz

    ;Itisclearthatthisalgebraissimpleanditcontains2a2.asafullsubalgebra ;Ourobjectivenowistoshowthattheabovetypedescribesal1finitesimpleOck

    ham

    ;algebrasWehavethefollowingresult

    ;Theorem5.Let(L;f)bedfinitesimpleOekhamalgebrawithnatom8.Thenthe ;structarv.olLisas|oflows:

    ;(1)Lhasnofixedpointsthenn?5oddandL2

    ;(2)L,?twofizedpointsthenisevenandL2

    ;(3)LhasuniquefizedpointthenLcontains2?n‖subalgebra.

    ;ProoLLeta?Lbeanatomandletmhethesmallestpositiveintegersuchthat ;f2m(n)=B.ByTheorem1,if0isneither1norafixedpointthenm>1Bythecorollary

    ;toTheorem3theelements.,,.(n),

    ,fm(n)areallatoms;andbythehypothesison

    ;mandthefactthatfisinjectivetheseatomsD,i-ealldistinctConsidertheelement

    ;n=nvf2(n)V…V,-2(n).

    ;Wehavef2()=n,whencewehavefrom_1,hemmalJthateither=1orisafixed ;point.SinceLhasatmosttwofixedpoint[,….weconsiderthefollowingcase~:

    ;f11Lhasnofixedpoints.Inthiscanenecessarily0=1andso1isajoinoftheatonls

    ;B,f2),,,m.(n).Itfollowsthatm=nandL2.ByTheorem3wethenhavethe ;situationofExample1withoddfnofixedpoints).

    ;(2)Lhastwofixedpoints.Inthiscane,by[1,Lemma1],therearecomplementaryin

    ;L.Theremustthereforeexistanatombthatdoesnotbelongtothesequencen,f2

(B),?,

    ;f2m2(q).IfPisthesmallestpositiveintegersuchthat,(6)=b,thenthesetofatomsof ;Lis

    ;————————.

    ;{n,,(n),.?,f2.,-2(n),b,f2(6),...f2p(6))

    ;

    ;218GHIN.ANNoFMATH.v01.17Ser.B ;andthefixedpointsare

    ;Tn1p--1

    ;n=

    ;V,.(.)and:V,‖(6).

    ;i=0t=0

    ;Sincea,arecomplementaryinLandsince,isadualautomorphism,foUowsthat

    ;[0,0d[l】墨【0,口】

    ;andhencethatp:m.Consequently,n=2m.Since1=aVisthejoinofa11theatoms,

    ;wehaveL2.ByTheorem3wethereforehavethesituationofExample1withev

    en

    ;(twofixedpoints).

    ;(3)Lhasuniquefixedpoint.IfLhasprecisely0Defixedpointthenn,f(?),

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