Hindman's Theorem, Ellis's Lemma, and Thompson's group $F$

TL;DR

This paper explores conjectural extensions of Hindman's Theorem and Ellis's Lemma to nonassociative systems, aiming to address the amenability of Thompson's group F, and presents partial results and analysis of these conjectures.

Contribution

It introduces new conjectural generalizations of classical theorems for nonassociative systems and relates them to the open problem of Thompson's group F's amenability.

Findings

Partial results supporting the conjectures

Analysis of the conjectures' implications

Connections to the amenability problem for Thompson's group F

Abstract

The purpose of this article is to formulate conjectural generalizations of Hindman's Theorem and Ellis's Lemma for nonassociative binary systems and relate them to the amenability problem for Thompson's group $F$. Partial results are obtained for both conjectures. The paper will also contain some general analysis of the conjectures.