Image partition regularity of matrices over commutative semigroups

Abstract

Let (S,+) be an infinite commutative semigroup with identity 0. Let u,v∈N and let A be a u×v matrix with nonnegative integer entries. If S is cancellative, let the entries of A come from Z. Then A is image partition regular over S (IPR/S) iff whenever S∖{0} is finitely colored, there exists x→∈(S∖{0})v such that the entries of Ax→ are monochromatic. The matrix A is centrally image partition regular over S (CIPR/S) iff whenever C is a central subset of S, there exists x→∈(S∖{0})v such that Ax→∈Cu. These notions have been extensively studied for subsemigroups of (R,+) or (R,⋅). We obtain some necessary and some sufficient conditions for A to be IPR/S or CIPR/S. For example, if G is an infinite divisible group, then A is CIPR/G iff A is IPR/Z. If for all c∈N, cS≠{0} and A is IPR/N, then A is IPR/S. If S is cancellative, c∈N, and cS={0}, we obtain a simple sufficient condition for A to be IPR/S. It is well-known that A is IPR/S if A is a first entries matrix with the property that cS is a central⁎ subset of S for every first entry c of A. We extend this theorem to first entries matrices whose first entries may not satisfy this condition. We discuss whether, if S is finitely colored, there exists x→∈(S∖{0})v, with distinct entries, for which the entries of Ax→ are monochromatic and distinct. Along the way, we obtain several new results about the algebra of βS, the Stone-Čech compactification of the discrete semigroup S.