Divisibility orders in $\beta N$

TL;DR

This paper explores five divisibility orders on the Stone-ch compactification of natural numbers, characterizing their structure and comparing their strength to the Rudin-Keisler order.

Contribution

It introduces and analyzes five divisibility orders on ch ch compactification, identifying their maximal, minimal elements, and their relation to Rudin-Keisler order.

Findings

Identified antichains of incompatible elements.

Characterized maximal and minimal elements.

Proved two divisibility orders are stronger than Rudin-Keisler order.

Abstract

We consider five divisibility orders on the Stone-\v{C}ech compactification $\beta N$. We find antichains of incompatible elements, and characterize maximal and minimal elements. The main result shows that two of these relations, $\mid_L$ and $\mid_{LN}$, are strictly stronger than the Rudin-Keisler order.