| Popis: |
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 ( I P R / S ) iff whenever S ∖ { 0 } is finitely colored, there exists x → ∈ ( S ∖ { 0 } ) v such that the entries of A x → are monochromatic. The matrix A is centrally image partition regular over S ( C I P R / S ) iff whenever C is a central subset of S, there exists x → ∈ ( S ∖ { 0 } ) v such that A x → ∈ C u . These notions have been extensively studied for subsemigroups of ( R , + ) or ( R , ⋅ ) . We obtain some necessary and some sufficient conditions for A to be I P R / S or C I P R / S . For example, if G is an infinite divisible group, then A is C I P R / G iff A is I P R / Z . If for all c ∈ N , c S ≠ { 0 } and A is I P R / N , then A is I P R / S . If S is cancellative, c ∈ N , and c S = { 0 } , we obtain a simple sufficient condition for A to be I P R / S . It is well-known that A is I P R / 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 A x → are monochromatic and distinct. Along the way, we obtain several new results about the algebra of βS, the Stone-Cech compactification of the discrete semigroup S. |
Full Text from ScienceDirect