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14
Preliminaries
101020Booleanalgebras
Booleanalgebracanbeintroducedasageneralizationofafieldofaset.
Considerasystem
A=(A,+,·,−,0,1),
whereAisanonemptyset,"+","·"arebinaryoperationsonA,"−"is
aunaryoperationonA,and0and1aresome(fixed)distinctelements
ofA.ThesystemAiscalledaBooleanalgebraifforarbitraryelements
ł,b,c∈Athefollowingaxioms(calledBooleanaxioms)aresatisfied
(1)ł+b=b+ł
(2)ł+(b+c)=(ł+b)+c
(3)ł·(b+c)=(ł·b)+(ł·c)
(4)ł+0=ł
(5)ł+(−ł)=1
ł·b=b·ł
ł·(b·c)=(ł·b)·c
ł+(b·c)=(ł+b)·(ł+c)
ł·1=ł
ł·(−ł)=0
Operations"+"and"·"arecalledBooleansumandBooleanproduct,
respectively.Operation"−"iscalledBooleancomplementation.0and1
arecalledBooleanzeroandBooleanunit,respectively.
Followingtheorderingonfields,thereisintroducedBooleanordering
onA:foranył,b∈A
ł⩽biffł+b=biffł·b=ł.
Therelation"⩽"partiallyordersA.
ThenexttheoremisessentialintheBooleanalgebratheory.
Theorem101(StoneRepresentationTheorem[41])EveryBooleanalge-
braisisomorphictoafieldofsets.
10103Quotientalgebras
LetAbeaBooleanalgebra.AnidealIonAisasubsetofAsuchthat
(1)0∈I,1̸∈I;
