Duality for image and kernel partition regularity of infinite matrices

TL;DR

This paper establishes a duality between image and kernel partition regularity for both finite and infinite matrices over Q, and explores its applicability to subsemigroups of Q.

Contribution

It introduces a duality theory linking image and kernel partition regularity for infinite matrices over Q, extending known results to broader contexts.

Findings

Duality holds for finite and infinite matrices over Q.

The duality extends to matrices over certain subsemigroups of Q.

Conditions under which the duality applies are characterized.

Abstract

A matrix A is image partition regular over Q provided that whenever Q - {0} is finitely coloured, there is a vector x with entries in Q - {0} such that the entries of Ax are monochromatic. It is kernel partition regular over Q provided that whenever Q - {0} is finitely coloured, the matrix has a monochromatic member of its kernel. We establish a duality for these notions valid for both finite and infinite matrices. We also investigate the extent to which this duality holds for matrices partition regular over proper subsemigroups of Q.