Ramsey Orderly Algebras as a New Approach to Ramsey Algebras

TL;DR

This paper introduces orderly algebras as a new framework for studying Ramsey algebras, establishing that an algebra is Ramsey if all its induced orderly algebras are Ramsey, thus providing a novel approach for further research.

Contribution

The paper proposes orderly algebras as a new concept and demonstrates their equivalence to Ramsey algebras, offering a fresh perspective for future investigations.

Findings

An algebra is Ramsey iff all its induced orderly algebras are Ramsey.

Orderly algebras provide a sound new approach to studying Ramsey algebras.

Abstract

Ramsey algebras are algebras that induce Ramsey spaces, which are generalizations of the Ellentuck space and Milliken's space. Previous work suggests a possible local version of Ramsey algebras induced by infinite sequences. Hence, we introduce a new structure called orderly algebra. Under our canonical setup, an algebra is Ramsey if and only if every of its induced orderly algebra is Ramsey. In this paper, we present justifications for this novel notion as a sound approach for further study on Ramsey algebras.