关于完备布尔代数的一点注解

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1、关于完备布尔代数的一点注解【摘要】在本研究中，给出了完备布尔代数是原子的等价刻画。同时，证明了一个完备的布尔代数是原子的当且仅当每一个真子代数是原子的。【关键词】完备布尔代数注解　　In[1],anatomlesscompleteBooleanalgebraiscalledsimpleifithasnoproperatomlesscompletesubalgebra,theequivalenceofwhichtorigidandminimalisprovedin[2].Asthesametime,thequestionofwetherasimplecompleteBooleanalgeb

2、raexistsisraisedfirstlyin[2].In[3],apositiveanswerisgiven.Similarly,inthenote,wewillgivesomeresultsofthecompleteatomicBooleanalgebraassameasthecompleteatomlessBooleanalgebra.　　ItiswellknownthatthepropertiesofcompleteBooleanalgebrascorrespondtopropertiesofgenericmodelsobtainedbyforcingwiththesealg

3、ebras,whichthemajorityworkoncompleteBooleanalgebracomefrom.Buttheidealofthisnotecomesfromlocaletheory,especially[4].6　　RecallthataframeoralocaleisacompletelatticeL,satisfyingtheinfinitedistributivelaw:a∈L,SLa∧∨S=∨{a∧s:s∈S}　　ByPt(L)wemeanthesetofprimeelementsofaframeL;aframeLissaidtobespatial,if

4、foranya∈L,a=∧{p∈Pt(A)

5、p≥a}.ThesubframeofFrameLisasubsetofit,whichisclosedunderfinitemeetsandarbitraryjoins;AsubsetofcompleteBooleanalgebrathatisclosedunderarbitrarymeetsandarbitraryjoinsiscalledcompletesubalgebra.AcompleteBooleanalgebraisaframe,thecompletesubalgebraofwhichisasubframe.Thepowerseto

6、fsetBisdenotedbyP(B);Let↑a={b∈L

7、b≥a}.Allmoreterminologyandnotationoflocaletheorywhichisnotexplainedhereistakenfrom[5];forgeneralbackgroundofcompleteBooleanalgebra,wereferto[6].　　MainResult　　Definition1AcompletelatticeLissaidtobegeneratedbyaset,ifthereexistsasubsetBofL,satisfyingthefollowingcondit

8、ions:①anytwoelementsofLisnotcompatible(a,b∈L,wehavenota≤b)②foranya∈L,a=∧6{b∈B

9、b≥a}③foranya∈L,b0∈↑a⌒B,a≠∧{b∈B

10、b≥a,b≠b0}Lemma2LetLbeacompletelattice.Lisisomorphictothepowersetofasetifandonlyifitisgeneratedbyaset.Proof:Itistrivially.SupposeLisgeneratedbyB.ForanySP(B),letf(S)=∧S.Bythedefinition1(2

11、),fissurjective.Inthefollowingweshowthatfisinjective.AssumingthereexistS1,S2∈P(B)witha=f(S1)=f(S2).IfS1≠S2,itisnoproblemtosupposethatthereexistsanelementb0∈S1suchthatb0S2andb0≥a,thena=∧{b∈B

12、b≥a,b≠b0}.Itiscontradictive