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The almost disjointness invariant for products of ideals. (English) Zbl 07628737

Summary: The almost disjointness numbers associated to the quotients determined by the transfinite products of the ideal of finite sets are investigated. A ZFC lower bound involving the minimum of the classical almost disjointness and splitting numbers is proved for these characteristics. En route, it is shown that the splitting numbers associated to these quotients are all equal to the classical splitting number. Finally, it is proved to be consistent that the almost disjointness numbers associated to these quotients are all equal to the second uncountable cardinal while the bounding number is the first uncountable cardinal. Several open problems are considered.

MSC:

03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
03E05 Other combinatorial set theory
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[1] Blass, A.; Dobrinen, N.; Raghavan, D., The next best thing to a P-point, J. Symb. Log., 80, 3, 866-900 (2015) · Zbl 1367.03078
[2] J. Brendle, Notes on cardinal characteristics of quotients, unpublished notes. Results communicated to first author in January 2021.
[3] Brendle, J., Mob families and mad families, Arch. Math. Log., 37, 3, 183-197 (1997) · Zbl 0905.03034
[4] Brendle, J.; Khomskii, Y., Mad families constructed from perfect almost disjoint families, J. Symb. Log., 78, 4, 1164-1180 (2013) · Zbl 1375.03057
[5] Brendle, J.; Raghavan, D., Bounding, splitting, and almost disjointness, Ann. Pure Appl. Log., 165, 2, 631-651 (2014) · Zbl 1405.03081
[6] Brendle, J.; Shelah, S., Ultrafilters on ω—their ideals and their cardinal characteristics, Trans. Am. Math. Soc., 351, 7, 2643-2674 (1999) · Zbl 0927.03073
[7] Grimeisen, G., Ein Approximationssatz für Bairesche Funktionen, Math. Ann., 146, 189-194 (1962) · Zbl 0133.00503
[8] Hernández-Hernández, F.; Hrušák, M., Cardinal invariants of analytic P-ideals, Can. J. Math., 59, 3, 575-595 (2007) · Zbl 1119.03046
[9] Katětov, M., On descriptive classification of functions, (General Topology and Its Relations to Modern Analysis and Algebra, III, Proc. Third Prague Topological Sympos.. General Topology and Its Relations to Modern Analysis and Algebra, III, Proc. Third Prague Topological Sympos., 1971 (1972)), 235-242 · Zbl 0309.54015
[10] Raghavan, D.; Shelah, S., Comparing the closed almost disjointness and dominating numbers, Fundam. Math., 217, 1, 73-81 (2012) · Zbl 1258.03058
[11] Schrittesser, D.; Törnquist, A.
[12] Shelah, S., On cardinal invariants of the continuum, (Axiomatic Set Theory. Axiomatic Set Theory, Boulder, CO, 1983. Axiomatic Set Theory. Axiomatic Set Theory, Boulder, CO, 1983, Contemp. Math., vol. 31 (1984), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 183-207 · Zbl 0583.03035
[13] Shelah, S., Two cardinal invariants of the continuum \((\mathfrak{d} < \mathfrak{a})\) and FS linearly ordered iterated forcing, Acta Math., 192, 2, 187-223 (2004) · Zbl 1106.03044
[14] Szymański, A.; Zhou, H. X., The behaviour of \(\omega^{2^\ast}\) under some consequences of Martin’s axiom, (General Topology and Its Relations to Modern Analysis and Algebra, V. General Topology and Its Relations to Modern Analysis and Algebra, V, Prague, 1981. General Topology and Its Relations to Modern Analysis and Algebra, V. General Topology and Its Relations to Modern Analysis and Algebra, V, Prague, 1981, Sigma Ser. Pure Math., vol. 3 (1983), Heldermann: Heldermann Berlin), 577-584 · Zbl 0506.54007
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