Properties of simple density ideals

Adam Kwela , Michał Popławski , Jarosław Swaczyna , Jacek Tryba

Abstract

Let G consist of all functions g : w -> [0, infinity) with g(n) -> infinity and n/g(n) negated right arrow 0. Then for each g is an element of G the family Z(g) = {A subset of w : lim(n ->infinity) card(A boolean AND n)/g(n) = 0} is an ideal associated to the notion of so-called upper density of weight g. Although those ideals have recently been extensively studied, they do not have their own name. In this paper, for Reader's convenience, we propose to call them simple density ideals. We partially answer [16, Problem 5.8] by showing that every simple density ideal satisfies the property from [16, Problem 5.8] (earlier the only known example was the ideal Z of sets of asymptotic density zero). We show that there are c many non-isomorphic (in fact even incomparable with respect to Katetov order) simple density ideals. Moreover, we prove that for a given A subset of G with card(A) < b one can construct a family of cardinality c of pairwise incomparable (with respect to inclusion) simple density ideals which additionally are incomparable with all Z(g) for g is an element of A. We show that this cannot be generalized to Katetov order as Z is maximal in the sense of Katetov order among all simple density ideals. We examine how many substantially different functions g can generate the same ideal Z(g) - it turns out that the answer is either 1 or c (depending on g).
Author Adam Kwela (FMPI / IM)
Adam Kwela,,
- Institute of Mathematics
, Michał Popławski
Michał Popławski,,
-
, Jarosław Swaczyna
Jarosław Swaczyna,,
-
, Jacek Tryba (FMPI / IM)
Jacek Tryba,,
- Institute of Mathematics
Journal seriesJournal of Mathematical Analysis and Applications, ISSN 0022-247X, e-ISSN 1096-0813, (N/A 70 pkt)
Issue year2019
Vol477
No1
Pages551-575
Publication size in sheets1.2
Keywords in Englishsimple density ideals, Erdős-Ulam ideals, Katětov order
ASJC Classification2603 Analysis; 2604 Applied Mathematics
DOIDOI:10.1016/j.jmaa.2019.04.046
URL https://doi.org/10.1016/j.jmaa.2019.04.046
Languageen angielski
Score (nominal)70
Score sourcejournalList
ScoreMinisterial score = 70.0, 05-02-2020, ArticleFromJournal
Publication indicators WoS Citations = 0; Scopus SNIP (Source Normalised Impact per Paper): 2018 = 1.187; WoS Impact Factor: 2018 = 1.188 (2) - 2018=1.219 (5)
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