Reply to Some Questions of Quotients when ultrafilters divide ultrafilters

TL;DR

This paper introduces a quotient-like operation on ultrafilters, explores self-divisible ultrafilters, and links algebraic stability with combinatorial properties like multiplicative delta sets.

Contribution

It formalizes the operation u/v on ultrafilters, characterizes self-divisible ultrafilters, and connects algebraic and combinatorial aspects of ultrafilter divisibility.

Findings

Characterization of self-divisible ultrafilters

Results on algebraic stability of multiplicative idempotents

Connection between multiplicative delta sets and self-divisible ultrafilters

Abstract

For ultrafilters u,v on N, the operation u/v is introduced and formalised which acts as quotient-like structures when v strongly divides u.Central to our study is the characterization of self-divisible ultrafilters in connection with the divisibility of u by u/v.Some results on the algebraic stability of multiplicative udempotents are presented.The paper also connects the combinatorial notions such as multiplicative delta sets,provoding characterization via self-divisible ultrafilters.