Density-like and generalized density ideals. (English) Zbl 07506777
Summary: 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.
MSC:
| 03E15 | Descriptive set theory |
| 11B05 | Density, gaps, topology |
| 28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
Keywords:
density-like ideal; lower semicontinuous submeasure; nonpathological exhaustive submeasure; generalized density idealReferences:
| [1] | Albiac, F. and Kalton, N. J., Topics in Banach Space Theory, second ed., Graduate Texts in Mathematics, vol. 233, Springer, Cham, 2016. · Zbl 1352.46002 |
| [2] | Balcerzak, M., Das, P., Filipczak, M., and Swaczyna, J., Generalized kinds of density and the associated ideals. Acta Mathematica Hungarica, vol. 147 (2015), no. 1, pp. 97-115. · Zbl 1374.03032 |
| [3] | Balcerzak, M. and Leonetti, P., On the relationship between ideal cluster points and ideal limit points. Topology and its Applications, vol. 252 (2019), pp. 178-190. · Zbl 1426.40003 |
| [4] | Borodulin-Nadzieja, P. and Farkas, B., Analytic P-ideals and Banach spaces. Journal of Functional Analysis, vol. 279 (2020), no. 8, Article no. 108702, p. 31. · Zbl 1484.03091 |
| [5] | Borodulin-Nadzieja, P., Farkas, B., and Plebanek, G., Representations of ideals in Polish groups and in Banach spaces , this Journal, vol. 80 (2015), no. 4, pp. 1268-1289. · Zbl 1436.03248 |
| [6] | Farah, I., Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Memoirs of the American Mathematical Society, vol. 148 (2000), no. 702. · Zbl 0966.03045 |
| [7] | Farah, I., How many Boolean algebras \({\ P\left(\mathbb{N}\right)/ \mathcal{I}\ }\) are there?Illinois Journal of Mathematics, vol. 46 (2002), no. 4, pp. 999-1033. · Zbl 1025.03050 |
| [8] | Farah, I., Examples of \(\epsilon \) -exhaustive pathological submeasures. Fundamenta Mathematicae, vol. 181 (2004), no. 3, pp. 257-272. · Zbl 1069.28002 |
| [9] | Farah, I., Analytic Hausdorff gaps. II. The density zero ideal. Israel Journal of Mathematics, vol. 154 (2006), pp. 235-246. · Zbl 1130.03032 |
| [10] | Filipów, R., Mrożek, N., Recław, I., and Szuca, P., Ideal convergence of bounded sequences , this Journal, vol. 72 (2007), no. 2, pp. 501-512. · Zbl 1123.40002 |
| [11] | Hrušák, M., Combinatorics of filters and ideals, Set Theory and Its Applications (L. Babinkostova, A. E. Caicedo, S. Geschke, and M. Scheepers, editors), Contemporary Mathematics, vol. 533, American Mathematical Society, Providence, RI, 2011, pp. 29-69. · Zbl 1239.03030 |
| [12] | Just, W. and Krawczyk, A., On certain Boolean algebras \({\ P\left(\omega \right) /\ I}\). Transactions of the American Mathematical Society, vol. 285 (1984), no. 1, pp. 411-429. · Zbl 0519.06011 |
| [13] | Kalton, N. J. and Roberts, J. W., Uniformly exhaustive submeasures and nearly additive set functions. Transactions of the American Mathematical Society, vol. 278 (1983), no. 2, 803-816. · Zbl 0524.28008 |
| [14] | Kwela, A., Erdős-Ulam ideals vs. simple density ideals. Journal of Mathematical Analysis and Applications, vol. 462 (2018), no. 1, pp. 114-130. · Zbl 1522.54004 |
| [15] | Kwela, A., Popławski, M., Swaczyna, J., and Tryba, J., Properties of simple density ideals. Journal of Mathematical Analysis and Applications, vol. 477 (2019), no. 1, pp. 551-575. · Zbl 1512.03060 |
| [16] | Leonetti, P., Limit points of subsequences. Topology and its Applications, vol. 263 (2019), pp. 221-229. · Zbl 1430.40002 |
| [17] | Louveau, A. and Veličković, B., A note on Borel equivalence relations. Proceedings of the American Mathematical Society, vol. 120 (1994), no. 1, pp. 255-259. · Zbl 0794.04002 |
| [18] | Louveau, A. and Veličković, B., Analytic ideals and cofinal types. Annals of Pure and Applied Logic, vol. 99 (1999), nos. 1-3, pp. 171-195. · Zbl 0934.03061 |
| [19] | Mátrai, T., More cofinal types of definable directed orders. Transactions of the American Mathematical Society, to appear. |
| [20] | Oliver, M. R., Continuum-many Boolean algebras of the form \({\ P\left(\omega \right) /\ \mathcal{I}} , \mathcal{I}\) Borel, this Journal, vol. 69 (2004), no. 3, pp. 799-816. · Zbl 1070.03030 |
| [21] | Schmeidler, D., Cores of exact games. I. Journal of Mathematical Analysis and Applications, vol. 40 (1972), pp. 214-225. · Zbl 0243.90071 |
| [22] | Solecki, S., Analytic ideals and their applications. Annals of Pure and Applied Logic, vol. 99 (1999), nos. 1-3, pp. 51-72. · Zbl 0932.03060 |
| [23] | Solecki, S., Tukey reduction among analytic directed orders. Zbornik Radova (Beograd), vol. 17 (2015), no. 25, pp. 209-220, Selected topics in combinatorial analysis. · Zbl 1474.03116 |
| [24] | Solecki, S. and Todorcevic, S., Cofinal types of topological directed orders. Annales de l’Institut Fourier, vol. 54 (2004), no. 6, 1877-1911. · Zbl 1071.03034 |
| [25] | Solecki, S. and Todorcevic, S., Avoiding families and Tukey functions on the nowhere-dense ideal. Journal of the Institute of Mathematics of Jussieu, vol. 10 (2011), no. 2, pp. 405-435. · Zbl 1221.03043 |
| [26] | Talagrand, M., Maharam’s problem. Annals of Mathematics. Second Series, vol. 168 (2008), no. 3, pp. 981-1009. · Zbl 1185.28002 |
| [27] | Uzcátegui Aylwin, C., Ideals on countable sets: a survey with questions. Revista Integración, Temas de Matemáticas, vol. 37 (2019), no. 1, pp. 167-198. · Zbl 1529.03241 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
