Volume 37 • Number 4 • 2014   • Ramsey Algebras and Strongly Reductible Ultrafilters Wen Chean Teh Abstract. Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman's Theorem. It is known that Martin's Axiom implies the existence of strongly summable ultrafilters, that is nonprincipal ultrafilters generated by sets of finite sums. Strongly reductible ultrafilters are analogues of strongly summable ultrafilters. Assuming Martin's Axiom, this paper shows the existence of nonprincipal strongly reductible ultrafilters for a nondegenerate Ramsey algebra. 2010 Mathematics Subject Classification: 03E50, 05D10 Full text: PDF