CENTRAL SETS IN COMMUTATIVE SEMIGROUPS AND PARTITION REGULARITY OF SYSTEMS OF LINEAR-EQUATIONS

Given a commutative semigroup (S, +) with identity 0 and u x v matrices A and B with nonnegative integers as entries, we show that if C = A - B satisfies Rado's columns condition over Z, then any central set in S contains solutions to the system of equations Ax half arrow pointing right = Bx half arrow pointing right. In particular, the system of equations Ax half arrow pointing right = Bx half arrow pointing right is then partition regular. Restricting our attention to the multiplicative semigroup of positive integers (so that coefficients become exponents) we show that the columns condition over Z is also necessary for the existence of solutions in any central set (while the distinct notion of the columns condition over Q is necessary and sufficient for partition regularity over N\{1}).