2021
DOI: 10.1017/jsl.2021.95
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Density-Like and Generalized Density Ideals

Abstract: We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega $ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal, vol. 80 (2015), pp. 1268–1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.

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Cited by 5 publications
(5 citation statements)
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“…Each of these classes has been extensively studied (e.g. [4,6,8,9,10,15,18,20]), and the relations between them were described in detail in [26]. We will present them in strictly increasing order, i.e.…”
Section: Preliminariesmentioning
confidence: 99%
See 1 more Smart Citation
“…Each of these classes has been extensively studied (e.g. [4,6,8,9,10,15,18,20]), and the relations between them were described in detail in [26]. We will present them in strictly increasing order, i.e.…”
Section: Preliminariesmentioning
confidence: 99%
“…Observe that matrix summability ideals are nonpathological analytic P-ideals as by [4,Proposition 12] every I(A) is equal to Exh(ϕ), where ϕ is given by ϕ(B) = sup n∈N k∈B a n,k for every B ⊆ N. There are plenty of nonpathological analytic P-ideals that are not matrix summability ideals, though, as can be seen by e.g. [20,Theorem 4.24]. Definition 2.14 ([10]).…”
Section: Preliminariesmentioning
confidence: 99%
“…An additional motivation comes from the fact that the study of ideals on countable sets and their representability may have some relevant potential for the study of the geometry of Banach spaces, see e.g. [6,7,23,25,29].…”
Section: Introductionmentioning
confidence: 99%
“…An additional motivation comes from the fact that that the study of ideals on countable sets and their representability may have some relevant potential for the study of the geometry of Banach spaces, see e.g. [6,7,24,27].…”
Section: Introductionmentioning
confidence: 99%