Image partition regularity of matrices over commutative semigroups

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}) such that the entries of A~x 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}) such that A~x ∈ C. 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 6= {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}), with distinct entries, for which the entries of A~x 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. Email addresses: [email protected] (Neil Hindman), [email protected] (Dona Strauss) URL: http://nhindman.us (Neil Hindman) 1This author acknowledges support received from the National Science Foundation (USA) via Grant DMS-1460023.