NDLI: $$\mathcal {F}$$ -finite embeddabilities of sets and ultrafilters

Content Provider	Springer Nature Link
Author	Luperi Baglini, Lorenzo
Copyright Year	2016
Abstract	Let S be a semigroup, let $$n\in \mathbb {N}$$ be a positive natural number, let $$A,B\subseteq S$$ , let $$\mathcal {U},\mathcal {V}\in \beta S$$ and let let $$\mathcal {F}\subseteq \{f:S^{n}\rightarrow S\}$$ . We say that A is $$\mathcal {F}$$ -finitely embeddable in B if for every finite set $$F\subseteq A$$ there is a function $$f\in \mathcal {F}$$ such that $$f\left( A^{n}\right) \subseteq B$$ , and we say that $$\mathcal {U}$$ is $$\mathcal {F}$$ -finitely embeddable in $$\mathcal {V}$$ if for every set $$B\in \mathcal {V}$$ there is a set $$A\in \mathcal {U}$$ such that A is $$\mathcal {F}$$ -finitely embeddable in B. We show that $$\mathcal {F}$$ -finite embeddabilities can be used to study certain combinatorial properties of sets and ultrafilters related with finite structures. We introduce the notions of set and of ultrafilter maximal for $$\mathcal {F}$$ -finite embeddability, whose existence is proved under very mild assumptions. Different choices of $$\mathcal {F}$$ can be used to characterize many combinatorially interesting sets/ultrafilters as maximal sets/ultrafilters, for example thick sets, AP-rich sets, $$\overline{K(\beta S)}$$ and so on. The set of maximal ultrafilters for $$\mathcal {F}$$ -finite embeddability can be characterized algebraically in terms of $$\mathcal {F}$$ . This property can be used to give an algebraic characterization of certain interesting sets of ultrafilters, such as the ultrafilters whose elements contain, respectively, arbitrarily long arithmetic, geoarithmetic or polynomial progressions. As a consequence of the connection between sets and ultrafilters maximal for $$\mathcal {F}$$ -finite embeddability we are able to prove a general result that entails, for example, that given a finite partition of a set that contains arbitrarily long geoarithmetic (resp. polynomial) progressions, one cell must contain arbitrarily long geoarithmetic (resp. polynomial) progressions. Finally we apply $$\mathcal {F}$$ -finite embeddabilities to study a few properties of homogeneous partition regular diophantine equations. Some of our results are based on connections between ultrafilters and nonstandard models of arithmetic.
Ending Page	734
Page Count	30
Starting Page	705
File Format	PDF
ISSN	09335846
e-ISSN	14320665
Journal	Archive for Mathematical Logic
Issue Number	5-6
Volume Number	55
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-05-24
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Nonstandard models in mathematics Algebra Ultrafilters Nonstandard analysis Semigroups Structure of topological semigroups Ramsey theory Mathematics Combinatorics Mathematical Logic and Foundations
Content Type	Text
Resource Type	Article
Subject	Philosophy Logic

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