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Idempotent filters and ultrafilters
[article]

Peter Krautzberger, Universitätsbibliothek Der FU Berlin, Universitätsbibliothek Der FU Berlin

2009

In this dissertation we will investigate filters on semigroups and their properties regarding algebra in the Stone-Cech compactification. The set of ultrafilters on a set S can be regarded as βS, the Stone-Cech compactification of S with the discrete topology. If S is a semigroup, we can define an associative operation on βS extending the operation on S. This can be done in such a way that the multiplication with a fixed right hand element is continuous; by the Ellis-Numakura Lemma there exist
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... dempotent elements in βS, i.e., idempotent ultrafilters. Idempotent ultrafilters play the central role in the field of algebra in the Stone-Cech compactification especially because they allow for elegant proofs of Ramsey-type theorems such as Hindman's Finite Sums Theorem, the Hales-Jewett Theorem and the Central Sets Theorem. Although ultrafilters are a natural topic of interest to set theorists, there are only few independence results regarding idempotent ultrafilters. In Chapter 3, we develop a uniform approach to adjoin idempotent ultrafilters by means of the forcing method. We are also able to produce a way to discern non-equivalent forcing notions by associating each forcing for adjoining idempotent ultrafilters with an already established notion for adjoining set theoretically interesting, non-idempotent ultrafilters. For these constructions, we study the notion of idempotent filter in Chapter 2. This notion is based on the natural generalization of the multiplication of ultrafilters to arbitrary filters. Idempotent filters are implicit in many applications in the field. Besides the usefulness for our forcing constructions, the notion of idempotent filter gives rise to a beautiful theory which we develop in Chapter 2. For example, idempotent filters correspond to subsemigroups of βS with strong closure properties and the notion is also a generalization of the concept of partial semigroups. The development of the theory of idempotent filters also allows us to give a simplified proof of Zelenyuk's Theorem on finite groups in the Stone- Cech c [...]

doi:10.17169/refubium-12831
fatcat:lgeomjgzfjfofaeuc6mmxnds7i