Exponentiations of ultrafilters

TL;DR

The paper addresses the partition regularity of exponential configurations on $\mathbb{N}$ by developing a framework for exponentiation of ultrafilters. It defines two noncommutative exponentiation operators, $E_{1}$ and $E_{2}$, on $\\beta\\mathbb{N}$ via $\\widehat{e_{1}}$ and $\\widehat{e_{2}}$, and studies their algebraic properties and interrelations with tensor products. A key result is the nonexistence of nonprincipal $E_{1}$-idempotents and partial nonexistence results for $E_{2}$-idempotents, highlighting the distinct interaction between additive, multiplicative, and exponential structures in $\\beta\\mathbb{N}$. The work raises multiple open questions about equalities among $E_{1}(p,q)$ and $E_{2}(p,q)$, the existence of $E_{2}$-idempotents, and the algebraic structure of exponentially rich ultrafilters, outlining promising directions for future research.

Abstract

In recent years, several problems regarding the partition regularity of exponential configurations have been studied in the literature, in some cases using the properties of specific ultrafilters. In this paper, we start to lay down the foundations of a general theory of exponentiations of ultrafilters, with a particular focus on their combinatorial properties and the existence of idempotents.

Theorems & Definitions (47)

• Definition 1.1
• Theorem 1.2: Additive Schur's Theorem
• Theorem 1.3: Multiplicative Schur's Theorem
• Theorem 1.4: Exponential Schur's Theorem
• Theorem 1.5: Van der Waerden's Theorem
• Definition 2.1
• Definition 2.4
• Lemma 2.5
• Theorem 2.6
• Theorem 2.7