Sticks and clubs

Fuchino, Sakaé; Shelah, Saharon; Soukup, Lajos

doi:10.1016/s0168-0072(97)00030-4

Cited by 35 publications

(51 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The second parts of (c) and (e) follow from Theorem 2.9. (Corollary 4.8) Results similar to Theorem 4.3 and Corollary 4.8 also hold for partial orderings with product-like structure as those considered in [9]. Thus, we can prove e.g.…”

Section: (D) This Follows Easily From (C)supporting

confidence: 62%

Coloring ordinals by reals

Brendle

Fuchino

2007

Fund. Math.

Self Cite

View full text Add to dashboard Cite

Abstract. We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ, λ) for regular κ > ℵ 1 and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals.These principles are strengthenings of C s (κ) and F s (κ) of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ 2 , ℵ 1 ) (hence also IP(ℵ 2 , ℵ 2 ) as well as HP(ℵ 2 )) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ 2 , ℵ 2 ) (hence also HP(ℵ 2 )) holds in a generic extension by countable support side-by-side product of Sacks or Prikry-Silver forcing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2).Relations between these principles and their influence on the values of the variations b ↑ , b h , b * , do of the bounding number b are studied. One of the consequences of HP(κ) besides C s (κ) is that there is no projective wellordering of length κ on any subset of ω ω. We construct a model in which there is no projective well-ordering of length ω 2 on any subset of ω ω (do = ℵ 1 in our terminology) while b * = ℵ 2 (Theorem 6.4).

show abstract

Section: (D) This Follows Easily From (C)supporting

confidence: 62%

Coloring ordinals by reals

Brendle

Fuchino

2007

Fund. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Unfortunately, as the following examples show, the situation for ideals is more complex than the situation for filters. (2). The following approach is closer in spirit to Example 3(2).…”

Section: ♠(F ♣ P(e) \ {∅} [ω 1 ] ℵ 1 ) Is the Same As ♣(E)mentioning

confidence: 97%

♣-like principles under CH

Just

2001

Fund. Math.

View full text Add to dashboard Cite

Abstract. Some relatives of the Juhász Club Principle are introduced and studied in the presence of CH. In particular, it is shown that a slight strengthening of this principle implies the existence of a Suslin tree in the presence of CH.Jensen's ♦-principle is an important strengthening of the Continuum Hypothesis that allows for a number of interesting constructions that cannot be carried out in ZFC + CH alone. Most notably, ♦ implies the existence of a Suslin tree. In [4], Ostaszewski introduced a weakening of ♦ known as ♣ that suffices for many of the constructions made possible by ♦. Devlin showed that ♣ + CH is equivalent to ♦ (see [4]). However, ♣ does not imply CH and is thus strictly weaker than ♦ (for an elegant proof, see [5], page 43). In [3], Juhász introduced a weakening of ♣ that will be denoted here by ♣ J . This principle is not equivalent to ♦ even under CH, and yet it retains some of the combinatorial power of the latter principle. Here we study some natural modifications of ♣ J . In particular, we show that some of these modifications, while still not equivalent to ♦ under CH, do imply the existence of a Suslin tree in the presence of CH.Throughout this note, letThe letter E will denote stationary subsets of S 0 , and S(E) will denote the family of all stationary subsets of E. Instead of S(S 0 ) we will write S. The family of all closed unbounded subsets of S 0 will be denoted by C. A ♣(E)-sequence is a sequence s α : α ∈ S 0 such that s α is a cofinal subset of order type ω for all α ∈ S 0 , and for each X ∈ [ω 1 ] ℵ 1 there exists α ∈ E with s α ⊂ X. The ♣(E)-principle asserts the existence of a ♣(E)-sequence. Instead of ♣(S 0 ) we simply write ♣.2000 Mathematics Subject Classification: 03E65, 03E35.

show abstract

“…In [10] Fuchino, Shelah, and Soukup introduced a new kind of side-by-side product of partial orders. Definition 5.6.…”

Section: The Pseudo-product Of Cohen Forcingsmentioning

confidence: 99%

On tightly ?-filtered Boolean algebras

Geschke¹

2002

Algebra Universalis

View full text Add to dashboard Cite

In this article we study the notion of tight κ-filteredness of a Boolean algebra for infinite regular cardinals κ. Tight ℵ 0 -filteredness is projectivity. We give characterizations of tightly κ-filtered Boolean algebras which generalize the internal characterizations of projectivity given by Haydon,Ščepin, and Koppelberg (see [15] or [17]). We show that for each κ there is an rc-filtered Boolean algebra which is not tightly κ-filtered. This generalizes a result ofŠčepin (see [15]). We prove that no complete Boolean algebra of size larger than ℵ 2 is tightly ℵ 1 -filtered. We give a new example of a model of set theory where P(ω) is tightly σ-filtered. We study the effect of the tight σ-filteredness of P(ω) on the automorphism group of P(ω)/f in. Introduction and plan of the paperKoppelberg [16] introduced and studied the notion of tight σ-filteredness of a Boolean algebra, which generalizes projectivity. Using this notion she gave uniform proofs of several mostly known results about the existence of certain homomorphisms into countably complete Boolean algebras. In this article we study tight κ-filteredness for all infinite regular cardinals κ. Koppelberg's tight σ-filteredness is tight ℵ 1 -filteredness. Projectivity is tight ℵ 0 -filteredness.Our research concerning tight κ-filteredness was initiated by a list of questions about tight σ-filteredness addressed by Fuchino. The first task was to obtain usable characterizations of tight κ-filteredness. These characterizations can be found in Section 2. Using these characterizations, in Section 3 we generalize some results of Koppelberg [17] on Stone spaces of projective Boolean algebras to Stone spaces of tightly κ-filtered Boolean algebras. Section 3 and the following sections are independent of each other, except for Section 6, which uses a result from Section 5. In Section 4 we show that for all infinite regular κ there are rc-filtered Boolean algebras that are not tightly κ-filtered. rc-filteredness is a generalization of projectivity Presented by M. Rubin.

show abstract

Sticks and clubs

Cited by 35 publications

References 9 publications

Coloring ordinals by reals

Coloring ordinals by reals

♣-like principles under CH

On tightly ?-filtered Boolean algebras

Contact Info

Product

Resources

About