2024/12/27 by Boris Šobot, Šobot, Boris
Computer Science · Mathematics · #03H15 #11U10 #54D35 #54D80 #Advanced Computational Techniques in Science and Engineering #Computational Geometry and Mesh Generation #Differential Equations and Numerical Methods #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2412.19753
openalex publication_date 2024/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A divisibility relation on ultrafilters on the set ℕ of natural numbers is defined as follows: \cal F\hspace1mm\widetilde|\hspace1mm\cal G if and only if every set in \cal F upward closed for divisibility also belongs to \cal G. Previously we isolated basic classes: powers of prime ultrafilters, and described the pattern of an ultrafilter, measuring the quantity of members of each basic class dividing a given ultrafilter. In this paper we define a topology on the set of basic classes which will allow us to calculate the pattern of the limit of a \widetilde|-increasing chain of ultrafilters. Using this we characterize which patterns can actually appear as patterns of an ultrafilter. Defining the =_∼-divisibility classes by identifying mutually divisible ultrafilters, in the respective quotient order (βℕ/=_∼,\widetilde|) we identify singleton classes and consider their patterns. Finally, we give a sufficient condition for a =_∼-divisibility class to have an immediate predecessor.