Cardinal sequences of length < ω₂ under GCH

István Juhász; Lajos Soukup; William Weiss

István Juhász ; Lajos Soukup ; William Weiss

Fundamenta Mathematicae, Tome 189 (2006), p. 35-52 / Harvested from The Polish Digital Mathematics Library

Résumé

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

Publié le : 2006-01-01

Zbl 1097.54004

EUDML-ID : urn:eudml:doc:282701

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     title = {Cardinal sequences of length < o2 under GCH},
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István Juhász; Lajos Soukup; William Weiss. Cardinal sequences of length < ω₂ under GCH. Fundamenta Mathematicae, Tome 189 (2006) pp. 35-52. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-1-3/