Sakaé Fuchino

Sticks and clubs

Abstract We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martins axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω 1 together with ¬CH and Martins axiom for countable p.o.-sets.

Partial orderings with the weak Freese-Nation property

Abstract A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping tf : P → [P] ⩽ℵ 0 such that, for any a, b ϵ P, if a ⩽ b then there exists c ϵ tf(a)∩tf(b) such that a ⩽ c ⩽ b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g. under CH, every ccc complete Boolean algebra has this property while, under b ⩾ ℵ 2 , there exists no complete Boolean algebra with the WFN (Theorem 6.2).

On a theorem of E. Helly

E. Helly’s theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.

On the weak Freese–Nation property of ?(ω)

Abstract. Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichońs diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models.

On the weak Freese-Nation property of complete Boolean algebras

Abstract The following results are proved: (a) In a model obtained by adding ℵ 2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □μ and cof ([μ] ℵ 0 ,⊆)=μ + hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of 〈 P (ω),⊆〉 is equivalent to the weak Freese-Nation property of any of C (κ) or R (κ) for uncountable κ. (d) Modulo the consistency of (ℵ ω+1 ,ℵ ω )↠(ℵ 1 ,ℵ 0 ) , it is consistent with GCH that C (ℵ ω ) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P (ω), ⊆ 〉 . These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.

On the Set-Generic Multiverse

The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver’s theorem and Bukovský’s theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory.

On L∞κ-free Boolean algebras

Abstract We study L ∞κ -freeness in the variety of Boolean algebras. It is shown that some of the theorems on L ∞κ -free algebras which are known to hold in varieties such as groups, abelian groups etc. are also true for Boolean algebras. But we also investigate properties such as the ccc of L ∞κ -free Boolean algebras which have no counterpart in the varieties above.

MODELS OF REAL-VALUED MEASURABILITY

Solovays random-real forcing ((1)) is the standard way of pro- ducing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure- theoretic properties of Solovays model that do not follow from the existence of real-valued measurability.

A game on partial orderings

Abstract We study the determinacy of the game G κ ( A ) introduced in Fuchino, Koppelberg and Shelah (to appear) for uncountable regular κ and several classes of partial orderings A . Among trees or Boolean algebras, we can always find an A such that G κ ( A ) is undetermined. For the class of linear orders, the existence of such A depends on the size of κ . In particular we obtain a characterization of κ = κ in terms of determinacy of the game G κ ( L ) for linear orders L .

Destructibility of stationary subsets of Pκλ

For a regular cardinal κ with κ<κ = κ and κ ≤ λ , we construct generically (forcing by a < κ-closed κ+-c. c. p. o.-set ℙ0) a subset S of {x ∈ Pκλ : x ∩ κ is a singular ordinal} such that S is stationary in a strong sense (FIAκλ -stationary in our terminology) but the stationarity of S can be destroyed by a κ+-c. c. forcing ℙ* (in Vℙ) which does not add any new element of Pκλ . Actually ℙ* can be chosen so that ℙ* is κ-strategically closed. However we show that such ℙ* itself cannot be κ-strategically closed or even <κ-strategically closed if κ is inaccessible. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)