关于完备布尔代数的一点注解论文

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1、关于完备布尔代数的一点注解论文【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.freellesspleteBooleanalgebraiscalledsimpleifithasnoproperatomlesspletesubalgebra,theequivalenceofinimalisprovedin2.Asthesametime,thequestionofplepleteBooleanalgebraexistsisraisedfirstlyin2.In3,apositiveansilarly,in

2、thenote,eresultsofthepleteatomicBooleanalgebraassameasthepleteatomlessBooleanalgebra.Itisodelsobtainedbyforcingajority.Buttheidealofthisnoteesfromlocaletheory,especially4.RecallthataframeoralocaleisapletelatticeL,satisfyingtheinfinitedistributivelaeanthesetofp

3、rimeelementsofaframeL;aframeLissaidtobespatial,ifforanya∈L,a=∧{p∈Pt(A)

4、p≥a}.ThesubframeofFrameLisasubsetofit,.freeleetsandarbitraryjoins;AsubsetofpleteBooleanalgebrathatisclosedunderarbitrarymeetsandarbitraryjoinsiscalledpletesubalgebra.ApleteBooleanalgebraisa

5、frame,thepletesubalgebraofe.Thepooreterminologyandnotationoflocaletheory5;forgeneralbackgroundofpleteBooleanalgebra,entsofLisnotpatible(a,b∈L,ma2LetLbeapletelattice.LisisomorphictothepoingthereexistS1,S2∈P(B)tosupposethatthereexistsanelementb0∈S1suchthatb0S2

6、andb0≥a,thena=∧{b∈B

7、b≥a,b≠b0}.Itiscontradictivetocondition(3)ofdefinition1.Lemma3LetLbeapleteBooleanalgebra.Foranya∈L,ifa=∧{p∈Pt(L)

8、p≥},thenforanyp0∈↑a⌒Pt(L),a≠∧{p∈Pt(L)

9、p≥a,p≠p0}.Proof:Leta=∧{p∈Pt(L)

10、p≥a,p≠p0}.ma4IfLisaspatialpleteBooleanalgebra,Lisgeneratedb

11、yPt(L).Proof:Lisspatial,thecondition1ofthedefinition1issatisfied.SinceLisaBooleanalgebra,theelementofPt(L)isacoatom,soanytentsPt(L)arenotpatible.Thecondition3canbeobtainedbylemma3.Theorem5IfLisapleteBooleanalgebra,thenthefolloic;(2)Lisspatial;(3)Lisgeneratedb

12、yaset;Proof:(1)(2)BythedualityofBooleanalgebras.(2)(3)Bythelemma4.(3)(1)Itistrivially.Remark:Someofthetheoremcanbefoundin6,hereentsaandpinLein4.Inthefolloportantrole.Lemma6Asubframeofspatialframeisspatial.Proof:SupposeLisaspatialframe,NisasubframeofL.Forany

13、p∈Pt(L),letp′=∨{b∈N

14、b≤p}.Notethatp′Pt(N),forassumingthatthereexisttentsx,y∈Noreover,x≤p′ory≤p′.Foranya∈N,icpleteBooleanalgebrahasnoatomiclesspletesubalgebra.Lemma8LetLbeanpleteBool