Ramsey Algebra and the Existence of Idempotent Ultrafilters

TL;DR

This paper develops a framework to study idempotent ultrafilters in Ramsey algebras, providing a positive partial answer to an open problem about their existence in such structures.

Contribution

It introduces a general framework for analyzing idempotent ultrafilters and proves their existence in certain countable Ramsey algebras, advancing understanding of their structure.

Findings

Every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some countable field of sets.

Addresses Carlson's open question about the existence of idempotent ultrafilters in Ramsey algebras.

Provides a positive result in the context of countable Ramsey algebras.

Abstract

Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper developes a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to a positive result that addresses Carlson's question in some way.