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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

;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).

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.

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,

).

;(b)Bvf(a)isafixedpoint.

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

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

;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.

;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.

;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

;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

;[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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