On the relationship between density and weak density in Boolean algebras

Bozeman, Kyle

doi:10.1090/s0002-9939-1991-1049841-5

Cited by 4 publications

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“…This function was introduced in Bozeman [2] where it is shown that hwd(A) is the minimum size of a set, X, such that for all a A + the set {a · x: x X} cannot be 2-reaped in A E a. Note in Bozeman [2] the terminology weakly dense is used instead of ''cannot be 2-reaped''. Also note that if a set is dense in A then this set is also hwd in A and so hwd(A) 5 y(A).…”

Section: Homogeneous Weak Densitymentioning

confidence: 99%

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Reaping numbers and operations on Boolean algebras

Peterson

1998

Algebra Universalis

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In this article we discuss the behaviour of reaping numbers under operations on Boolean algebras. Some major results are as follows.(1) If I is infinite, thenIf L is a linear order with a first element, then hwd(Intalg(L))= d(L). (5) If A is the completion of an interval algebra, then hwd(A)= y(A). (6) If T is a tree with no maximal elements, then hwd(Treealg(T))= T. For the notation here, see below.

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Section: Homogeneous Weak Densitymentioning

confidence: 99%

“…By lemma 6.3 we have hwd(A) ] c(A) for complete BA's. In Bozeman [2] it is shown that if A is complete and homogeneous for the functions y and hwd then r 2 (A)= hwd(A). He also shows, assuming GCH, that if A is complete and homogeneous in y and hwd then y(A) = hwd(A).…”

Section: Lemma 64 Suppose a Is An Infinite Ba And X ¤ A Such That Xmentioning

confidence: 99%

Reaping numbers and operations on Boolean algebras

Peterson

1998

Algebra Universalis

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“…This function has been extensively studied in the general context of Boolean algebras. See, for example, Balcar and Simon [1992], Peterson [1998], Bozeman [1991], Monk [1996].…”

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confidence: 99%

Continuum cardinals generalized to Boolean algebras

Monk¹

2001

J. symb. log.

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A number of specific cardinal numbers have been defined in terms of /fin or ωω. Some have been generalized to higher cardinals, and some even to arbitrary Boolean algebras. Here we study eight of these cardinals, defining their generalizations to higher cardinals, and then defining them for Boolean algebras. We then attempt to completely describe their relationships within each of several important classes of Boolean algebras.The generalizations to higher cardinals might involve several cardinals instead of just one as in the case of ω, For example, the number a associated with maximal almost disjoint families of infinite sets of integers can be generalized to talk about maximal subsets of [κ]μ subject to the pairwise intersections having size less than ν. (For this multiple generalization of . see Monk [2001].) For brevity we do not consider such generalizations, restricting ourselves to just one cardinal. The set-theoretic generalizations then associate with each infinite cardinal κ some other cardinal λ, defined as the minimum of cardinals with a certain property.The generalizations to Boolean algebras assign to each Boolean algebra some cardinal λ, also defined as the minimum of cardinals with a certain property.For the theory of the original “continuum” cardinal numbers, see Douwen [1984]. Balcar and Simon [1989]. and Vaughan [1990].I am grateful to Mati Rubin for some conversations concerning these functions for superatomic algebras, and to Bohuslav Balcar for information concerning the function h.The notation for set theory is standard. For Boolean algebras we follow Koppelberg [1989], but recall at the appropriate place any somewhat unusual notation.

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Cardinal Invariants on Boolean Algebras

Monk

2014

Progress in Mathematics

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The purpose of these notes is to describe the progress made on the 97 open problems formulated in the book Cardinal invariants on Boolean algebras, hereafter denoted by [CI]. Although we assume acquaintance with that book, we give some background for many problems, and state without proof most results relevant to the problems, as far as the author knows. Many of the problems have been solved, at least partially, by Saharon Shelah. For the unpublished papers of Shelah mentioned in the references, see his archive, which can currently be obtained at the URL http://shelah.logic.at/ We follow the notation of Koppelberg [89] and of [CI]. All Boolean algebras are assumed to be infinite, unless otherwise indicated.At this time, the problems fully solved are:

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On the relationship between density and weak density in Boolean algebras

Abstract: Abstract.Given a homogeneous, complete Boolean algebra B, it is shown that d(B) < min(2

Cited by 4 publications

References 3 publications

Reaping numbers and operations on Boolean algebras

Reaping numbers and operations on Boolean algebras

Continuum cardinals generalized to Boolean algebras

Cardinal Invariants on Boolean Algebras

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