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Chapter1
15
(2)ifł,b∈Ithenł+b∈I;
(3)ifł,b∈A,ł∈Iandb⩽łthenb∈I.
Thereisarelationbetweenidealsandhomomorphisms.LetAandB
beBooleanalgebrasandleth:A→Bbeahomomorphism.Then
I={ł∈A:h(ł)=0}
isanidealonA.
LetIbeanidealonA.Considertheequivalencerelation∼onA,for
allł,b∈A
ł∼biffł△b∈I,
whereł△b=(ł−b)+(b−ł).LetBbethesetofallequivalenceclasses
[ł]forł∈A,where
[ł]={b∈A:b∼ł},
endowedwiththefollowingoperations
[ł]+[b]=[ł+b]
[ł]·[b]=[ł·b]
−[ł]=[−ł]
0=[0]
1=[1]
ThenBisBooleanalgebraandisanisomorphicimageofAunderthe
assumptionh(ł)=[ł].BiscalledaquotientalgebraanddenotedbyA/I.
10104Treeideals
LetK⊂ωbeaset(finiteorinfinite).AsetT⊂K<ωiscalledatreeif
t|n∈Tforallt∈Tandn⩽|t|(i.e.,Tiscloseddownwardsunderinitial
segments).Itisassumedthattreeshavenoterminalnodes.
LetTmeansafamilyofalltrees.ForeachT∈Tandt∈Ttheset
split(t,T)=|{n∈K:t
⌢n∈T}|
denotesthenumberofsuccessorsofnodesinT.
AtreeTiscalled
1.Sackstree(orperfecttree)ifK={0,1}andforeacht∈Tthereis
ans∈Tsuchthatt⊂sandsplit(t,T)=2;
