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# Journal of Combinatorics

## Volume 8 (2017)

### Number 4

### Duality for image and kernel partition regularity of infinite matrices

Pages: 653 – 672

DOI: http://dx.doi.org/10.4310/JOC.2017.v8.n4.a5

#### Authors

#### Abstract

A matrix $A$ is *image partition regular* over $\mathbb{Q}$ provided that whenever $\mathbb{Q} \backslash \lbrace 0 \rbrace$ is finitely coloured, there is a vector $\vec{x}$ with entries in $\mathbb{Q} \backslash \lbrace 0 \rbrace$ such that the entries of $A\vec{x}$ are monochromatic. It is *kernel partition regular over* $\mathbb{Q}$ provided that whenever $\mathbb{Q} \backslash \lbrace 0 \rbrace$ 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 $\mathbb{Q}$.

#### Keywords

kernel partition regularity, image partition regularity, infinite matrices

#### 2010 Mathematics Subject Classification

05D10

N. Hindman acknowledges support received from the National Science Foundation via Grants DMS-1160566 and DMS-1460023.

Received 10 September 2015

Published 17 July 2017