Abstract
We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations \(\mid _M\) and \({\widetilde{\mid }}\). The set of its minimal elements proves to be very rich, and the \({\widetilde{\mid }}\)-hierarchy is used to get a better intuition of this richness. We find the place of the set of \({\widetilde{\mid }}\)-maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets of \({\mathbb {N}}\), and compare it to other such notions, important for infinite combinatorics and topological dynamics.
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Funding
The author gratefully acknowledges financial support of the Science Fund of the Republic of Serbia (call PROMIS, project CLOUDS, Grant no. 6062228) and Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant no. 451-03-9/2021-14/200125).
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The author gratefully acknowledges financial support of the Science Fund of the Republic of Serbia (call PROMIS, project CLOUDS, grant no. 6062228) and Ministry of Education, Science and Technological Development of the Republic of Serbia (grant no. 451-03-9/2021-14/200125). The author wishes to thank the referee for careful reading of the manuscript, and in particular for providing a solution for one of the open problems.
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Šobot, B. Multiplicative finite embeddability vs divisibility of ultrafilters. Arch. Math. Logic 61, 535–553 (2022). https://doi.org/10.1007/s00153-021-00799-y
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DOI: https://doi.org/10.1007/s00153-021-00799-y