Duality for image and kernel partition regularity of infinite matrices

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 A x 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.

Recent Results on Partition Regularity of Infinite Matrices

“…There is a corresponding result starting with an image partition regular matrix, but it only applies when S = Q. In fact, it is shown in [16] that if S is a nontrivial proper subsemigroup of Q + = {x ∈ Q : x > 0} or a nontrivial proper subgroup of Q, then there is a 3 × 2 matrix B which is IPR/S for which the conclusion of Theorem 3.2 fails.…”