In this paper we study a notion of preorder that arises in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-Cech compactification of the discrete space of natural ˇ numbers. In particular, we prove that there exist ultrafilters maximal for finite embeddability, and we show that the set of such ultrafilters is the closure of the minimal bilateral ideal in the semigroup (βN, ⊕), namely K(βN, ⊕). By combining this characterization with some known combinatorial properties of certain families of sets we easily derive some combinatorial properties of ultrafilters in K(βN, ⊕). We also give an alternative proof of our main result based on nonstandard models of arithmetic.
Ultrafilters maximal for finite embeddability / L. Luperi Baglini. - In: JOURNAL OF LOGIC AND ANALYSIS. - ISSN 1759-9008. - 6:(2014 Oct), pp. 6.1-6.16. [10.4115/jla.2014.6.6]
Ultrafilters maximal for finite embeddability
L. Luperi Baglini
2014
Abstract
In this paper we study a notion of preorder that arises in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-Cech compactification of the discrete space of natural ˇ numbers. In particular, we prove that there exist ultrafilters maximal for finite embeddability, and we show that the set of such ultrafilters is the closure of the minimal bilateral ideal in the semigroup (βN, ⊕), namely K(βN, ⊕). By combining this characterization with some known combinatorial properties of certain families of sets we easily derive some combinatorial properties of ultrafilters in K(βN, ⊕). We also give an alternative proof of our main result based on nonstandard models of arithmetic.File | Dimensione | Formato | |
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