Minimally generated Bo olean algeb ras Piotr Bo ro - - PDF document

▶
minimally generated bo olean algeb ras piotr bo ro dulin
SMART_READER_LITE
LIVE PREVIEW

Minimally generated Bo olean algeb ras Piotr Bo ro - - PDF document

Sep 15, 2023 380 likes •599 views

Minimally generated Bo olean algeb ras Piotr Bo ro dulin-Nadzieja Inst ytut Matemat yczny Uniw ersytetu W ro c la wskiego Denition A Bo olean algeb ra B extends an algeb ra A minimally if A B and there

slide-1
SLIDE 1 Minimally generated Bo
  • lean
algeb ras Piotr Bo ro dulin-Nadzieja Inst ytut Matemat yczny Uniw ersytetu W ro c la wskiego
slide-2
SLIDE 2 Denition A Bo
  • lean
algeb ra B extends an algeb ra A minimally if A
  • B
and there is no algeb ra C such that A ( C ( B .
slide-3
SLIDE 3 Denition A Bo
  • lean
algeb ra B extends an algeb ra A minimally if A
  • B
and there is no algeb ra C such that A ( C ( B . Denition A Bo
  • lean
algeb ra A is minimally generated if there is a sequence (A
  • )
< such that
  • A
= f0 ; 1 g;
  • A
+1 extends A
  • minimally
fo r
  • <
;
  • A
  • =
S < A
  • fo
r limit
  • ;
  • A
= S < A
  • .
slide-4
SLIDE 4 Denition A Bo
  • lean
algeb ra A is minimally generated if there is a sequence (A
  • )
< such that
  • A
= f0 ; 1 g;
  • A
+1 extends A
  • minimally
fo r
  • <
;
  • A
  • =
S < A
  • fo
r limit
  • ;
  • A
= S < A
  • .
A top
  • logical
space K is minimally generated if it is a Stone space
  • f
a minimally generated Bo
  • lean
algeb ra.
slide-5
SLIDE 5 Denition A Bo
  • lean
algeb ra A is small if it do es not contain an uncountable indep endent family .
slide-6
SLIDE 6 Denition A Bo
  • lean
algeb ra A is small if it do es not contain an uncountable indep endent family . Theo rem (Kopp elb erg) Minimally generated Bo
  • lean
algeb ras a re small.
slide-7
SLIDE 7 Denition A Bo
  • lean
algeb ra A is small if it do es not contain an uncountable indep endent family . Theo rem (Kopp elb erg) Minimally generated Bo
  • lean
algeb ras a re small. Theo rem (Kopp elb erg) There is a small Bo
  • lean
algeb ra which is not minimally generated.
slide-8
SLIDE 8 F act Every (nitely additive) measure
  • n
a Bo
  • lean
algeb ra A can b e uniquely extended to a Radon measure
  • n
the Stone space
  • f
A .
slide-9
SLIDE 9 F act Every (nitely additive) measure
  • n
a Bo
  • lean
algeb ra A can b e uniquely extended to a Radon measure
  • n
the Stone space
  • f
A . Denition A measure
  • dened
  • n
a Bo
  • lean
algeb ra A is sepa rable if L 1 ( ) is sepa rable.
slide-10
SLIDE 10 F act Every (nitely additive) measure
  • n
a Bo
  • lean
algeb ra A can b e uniquely extended to a Radon measure
  • n
the Stone space
  • f
A . Denition A measure
  • dened
  • n
a Bo
  • lean
algeb ra A is sepa rable if L 1 ( ) is sepa rable. F act Every big Bo
  • lean
algeb ra ca rries a non- sepa rable measure.
slide-11
SLIDE 11 F act Every (nitely additive) measure
  • n
a Bo
  • lean
algeb ra A can b e uniquely extended to a Radon measure
  • n
the Stone space
  • f
A . Denition A measure
  • dened
  • n
a Bo
  • lean
algeb ra A is sepa rable if L 1 ( ) is sepa rable. F act Every big Bo
  • lean
algeb ra ca rries a non- sepa rable measure. Theo rem (F remlin) M A (! 1 ) implies that small Bo
  • lean
algeb ras admit
  • nly
sepa rable mea- sures.
slide-12
SLIDE 12 Theo rem Minimally generated Bo
  • lean
alge- b ras admit
  • nly
sepa rable measures.
slide-13
SLIDE 13 Theo rem Minimally generated Bo
  • lean
alge- b ras admit
  • nly
sepa rable measures. Mo re p recisely: If
  • is
a measure
  • n
a minimally generated Bo
  • lean
algeb ra then it is a count- able sum
  • f
w eakly unifo rmly regula r measures.
slide-14
SLIDE 14 Co rolla ry: The follo wing classes
  • f
Bo
  • lean
algeb ras (spaces) admit
  • nly
sepa rable measures:
  • interval
algeb ras (o rdered spaces);
  • tree
algeb ras;
  • sup
eratomic algeb ras (scattered spaces);
  • monotonically
no rmal spaces.
slide-15
SLIDE 15 Co rolla ry: The follo wing classes
  • f
Bo
  • lean
algeb ras (spaces) a re not included (at least, not in ZF C) in the class
  • f
minimally generated Bo
  • lean
algeb ras:
  • retractive
algeb ras;
  • Co
rson compacta.
slide-16
SLIDE 16 Another co rolla ries: A length
  • f
minimally generated Bo
  • lean
alge- b ra has to b e a limit
  • rdinal
but not necessa rily a ca rdinal. In the realm
  • f
minimally generated spaces ccc = sepa rabilit y Mo reover, if w e assume Suslin Conjecture then ccc = existence
  • f
strictly p
  • sitive
measure
slide-17
SLIDE 17 Emov Problem Is there a compact innite space K such that
  • K
do es not contain a nontrivial convergent sequence;
  • K
do es not contain a cop y
  • f
  • !
?
slide-18
SLIDE 18 Emov Problem Is there a compact innite space K such that
  • K
do es not contain a nontrivial convergent sequence;
  • K
do es not contain a cop y
  • f
  • !
? Theo rem (F edo rchuk) CH implies the exis- tence
  • f
such space. It is not kno wn if it is consistent to assume that there is no Emov space.
slide-19
SLIDE 19 Problem Ho w to cha racterize minimally gen- erated spaces? Theo rem (Kopp elb erg) Every minimally gen- erated space has a tree
  • base.
slide-20
SLIDE 20 Problem Ho w to cha racterize minimally gen- erated spaces? Theo rem (Kopp elb erg) Every minimally gen- erated space has a tree
  • base.
Question Is it true that every minimally gen- erated space is discretely generated?