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Cardinal invariants of ultraproducts of Boolean algebras

Andrzej Rosłanowski, Saharon Shelah (1998)

Fundamenta Mathematicae

We deal with some problems posed by Monk [Mo 1], [Mo 3] and related to cardinal invariants of ultraproducts of Boolean algebras. We also introduce and investigate several new cardinal invariants.

Cardinal sequences and Cohen real extensions

István Juhász, Saharon Shelah, Lajos Soukup, Zoltán Szentmiklóssy (2004)

Fundamenta Mathematicae

We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most ( 2 ) V levels of size ω. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of regular and of 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.

Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, William Weiss (2006)

Fundamenta Mathematicae

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put λ ( α ) = s ( α ) : s ( 0 ) = λ = m i n [ s ( β ) : β < α ] . We show that f ∈ (α) iff for some natural number n there are infinite cardinals λ i > λ > . . . > λ n - 1 and ordinals α , . . . , α n - 1 such that α = α + + α n - 1 and f = f f . . . f n - 1 where each f i λ i ( α i ) . Under GCH we prove that if α < ω₂ then (i) ω ( α ) = s α ω , ω : s ( 0 ) = ω ; (ii) if λ > cf(λ) = ω, λ ( α ) = s α λ , λ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d i n α ; (iii) if cf(λ) = ω₁, λ ( α ) = s α λ , λ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d a n d s u c c e s s o r - c l o s e d i n α ; (iv) if cf(λ) > ω₁, λ ( α ) = α λ . This yields a complete characterization of the classes (α) for all α < ω₂,...

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