Maximal chains in atomic Boolean algebras

Day, George W.

doi:10.4064/fm-67-3-293-296

Cited by 11 publications

(7 citation statements)

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“…First we are going to find isomorphic copies of maximal chains of the complete Boolean algebra P(ω) in the unit interval. We will use the following, general result of Day (see [3]). …”

Section: Types Of Maximal Chains In Similar Posetsmentioning

confidence: 99%

“…Types of maximal chains in several posets were considered by many authors. 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.…”

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confidence: 99%

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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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“…First we are going to find isomorphic copies of maximal chains of the complete Boolean algebra P(ω) in the unit interval. We will use the following, general result of Day (see [3]). …”

Section: Types Of Maximal Chains In Similar Posetsmentioning

confidence: 99%

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confidence: 99%

Maximal Chains in Positive Subfamilies of P(ω)

Kurilić

2011

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“…Maximal chains in Boolean algebras have been discussed before in Day [3], Jakubík [5], and Koppelberg [7]. It turns out that an important role is played here by saturated Boolean algebras (in the model-theoretic sense).…”

Section: Maximal Chainsmentioning

confidence: 99%

Towers and maximal chains in Boolean algebras

Monk

2007

Algebra univers.

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A tower in a Boolean algebra A is a strictly increasing sequence, of regular length, of elements of A with supremum 1. We consider the following functions:These are called the spectrum of towers of A and the spectrum of maximal chains of A, respectively. Our main results are (1) for any nonempty set K of regular cardinals there is an atomless Boolean algebra A such that tspect(A) = K; (2) under GCH, an analogous result holds for maximal chains. Note that towers do not exist in some Boolean algebras, for example in the finite-cofinite algebra on an uncountable set.

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“…1 Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia. e-mail: milos@dmi.uns.ac.rs 2 Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11001 Belgrade, Serbia. e-mail: borisa@mi.sanu.ac.rs So, for example, there are maximal chains of copies of the random poset isomorphic to (0, 1], to the Cantor set without 0, and to α * , for each countable limit ordinal α.…”

Section: Introductionmentioning

confidence: 99%