Towers and maximal chains in Boolean algebras

Monk, J. Donald

doi:10.1007/s00012-007-2002-8

Cited by 8 publications

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“…We mention the characterizations in power-set algebras of Kuratowski [8]; in atomic Boolean algebras of Day [3]; in the interval algebra of [0, 1) R of Koppelberg [6] and in the poset E(Q) of elementary submodels of the rational line (see [9]). We note that the papers [10] of McKenzie and Monk and [11] of Monk contain an extensive analysis of chains in Boolean algebras. Section 2 contains a preliminary analysis of positive families.…”

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confidence: 99%

Maximal Chains in Positive Subfamilies of P(ω)

Kurilić

2011

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A family P ⊂ [ω] ω is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/ Fin. For example, if I ⊂ P(ω) is an ideal containing the ideal Fin of finite subsets of ω, then P(ω) \ I is a positive family and the set Dense(Q) of dense subsets of the rational line is a positive family which is not the complement of some ideal on P(Q). We prove that, for a positive family P, the order types of maximal chains in the complete lattice P ∪ {∅}, ⊂ are exactly the order types of compact nowhere dense subsets of the real line having the minimum nonisolated. Also we compare this result with the corresponding results concerning maximal chains in the Boolean algebras P(ω) and Intalg[0, 1) R and the poset E(Q), where E(Q) is the set of elementary submodels of the rational line.

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confidence: 99%

Maximal Chains in Positive Subfamilies of P(ω)

Kurilić

2011

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“…The functions considered in this paper are as follows. The functions c and πχ inf are discussed in 5; s mm in 8; \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\frak l}$\end{document} in 7; \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\frak f}$\end{document} in 9; the others in 6.…”

Section: Notationmentioning

confidence: 99%

“…Maximal chains in Boolean algebras were investigated in 7, but the following simple connection with the well‐known tower number was not mentioned.…”

Section: Arbitrary Atomless Boolean Algebrasmentioning

confidence: 99%

Remarks on continuum cardinals on Boolean algebras

Monk

2012

Mathematical Logic Qtrly

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“…We restrict ourselves here to atomless BAs. Towers in atomless Boolean algebras are discussed in Monk [11] and Monk [12]. …”

Section: Towersmentioning

confidence: 99%

“…This notion is discussed in Monk [12]. Lemma 9.1 If X is a maximal chain in A, with A an atomless BA, then X is a maximal chain in A ⊕ B.…”

Section: Small Lengthmentioning

confidence: 99%

Special subalgebras of Boolean algebras

Monk

2010

Math. Log. Quart.

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Key words Subalgebras of Boolean algebras, cardinal functions. MSC (2000) 06E05, 03G05We consider eight special kinds of subalgebras of Boolean algebras. In Section 1 we describe the relationships between these subalgebra notions. In succeeding sections we consider how the subalgebra notions behave with respect to the most common cardinal functions on Boolean algebras. Definitions and relationshipsIn notation we follow Koppelberg [4] and Monk [9], but also we give key definitions that we will work with.AThis is illustrated by the following diagram:This is the definition of projective subalgebra given in Koppelberg [5, p. 752].A ≤ rc B iff A is a relatively complete subalgebra of B; see Koppelberg [4, p. 123]. This means that for each b ∈ B there is a smallest a ∈ A with b ≤ a, and also a greatestA ≤ s B iff B is a simple extension of A. This means that B is obtained from A by adding just one element, and all elements then necessary to form a BA; we denote B by A(x), where x is the new element.A ≤ m B iff B is a minimal extension of A. Thus B is a simple extension of A and there is no BA C such thatA ≤ mg B iff B is a minimally generated extension of A. Thus B can be obtained from A by a sequence of minimal extensions, with unions taken at limit steps.A ≤ π B iff A is a dense subalgebra of B. Here dense means that for every b ∈ B + there is an a ∈ A + such that a ≤ b. *

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Towers and maximal chains in Boolean algebras

Cited by 8 publications

References 11 publications

Maximal Chains in Positive Subfamilies of P(ω)

Maximal Chains in Positive Subfamilies of P(ω)

Remarks on continuum cardinals on Boolean algebras

Special subalgebras of Boolean algebras

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