2024/01/04 by Boris Šobot, Šobot, Boris · 1 citation
Computer Science · Mathematics · #03H15 #11U10 #54D35 #54D80 #Advanced Algebra and Logic #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2401.02302
openalex publication_date 2024/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An extension of the divisibility relation on ℕ to the set βℕ of ultrafilters on ℕ was defined and investigated in several papers during the last ten years. Here we make a survey of results obtained so far, adding several results connecting the themes of different stages of the research. The highlights include: separation of βℕ into the lower part L (with its division into levels) and the upper part; identifying basic ingredients (powers of primes) and fragmentation of each ultrafilter into them; finding the corresponding upward closed sets belonging to an ultrafilter with given basic ingredients; existence and number of successors and predecessors of a given divisibility class; extending the congruence relation (in two ways) and checking properties of the obtained relations.