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Archive for Mathematical Logic

, Volume 55, Issue 3–4, pp 475–491 | Cite as

Ramsey algebras and the existence of idempotent ultrafilters

  • Wen Chean TehEmail author
Article

Abstract

Hindman’s Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman’s Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper develops a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to a positive result that addresses Carlson’s question in some way.

Keywords

Ramsey algebra Hindman’s Theorem Idempotent ultrafilter Strongly reductible ultrafilter 

Mathematics Subject Classification

05A17 05D10 03E05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversiti Sains Malaysia (USM)PenangMalaysia

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