Finite Embeddability of Sets and Ultrafilters

Blass, Andreas; Nasso, Mauro Di

doi:10.4064/ba8024-1-2016

Cited by 9 publications

(30 citation statements)

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“…A similar relation between sets of natural numbers, named "finite embeddability", has been considered in [17, §4] (see also [9], where that notion was extended to ultrafilters). The difference is that "A finitely embedded in B" only requires the inclusion x + (A ∩ I) ⊆ B ∩ (x + I).…”

Section: Hyper-shiftsmentioning

confidence: 99%

“…Partition regularity of (nonlinear) polynomial equations by nonstandard methods is the subject-matter of the paper [36]. In [9], a notion of finite embeddability between sets and between ultrafilters is investigated, also with the use of the hyper-shifts of §5. The papers [37,38] continue that line of research: the nonstandard approach is exploited to further investigating the relationships between finite embeddability relations, algebraic properties in (βN, ⊕), and combinatorial structure of sets of natural numbers.…”

Section: Final Remarks and Open Questionsmentioning

confidence: 99%

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Hypernatural Numbers as Ultrafilters

Nasso

2015

Nonstandard Analysis for the Working Mathematician

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Abstract. In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers. Introduction.Ultrafilters are really peculiar and multifaced mathematical objects, whose study turned out a fascinating and often elusive subject. Researchers may have diverse intuitions about ultrafilters, but they seem to agree on the subtlety of this concept; e.g., read the following quotations: "The space βω is a monster having three heads" (J. van Mill [41]); ". . . the somewhat esoteric, but fascinating and very useful objectThe notion of ultrafilter can be formulated in diverse languages of mathematics: in set theory, ultrafilters are maximal families of sets that are closed under supersets and intersections; in measure theory, they are described as {0, 1}-valued finitely additive measures defined on the family of all subsets of a given space; in algebra, they exactly correspond to maximal ideals in rings of functions F I where I is a set and F is a field. Ultrafilters and the corresponding construction of ultraproduct are a common tool in mathematical logic, but they also have many applications in other fields of mathematics, most notably in topology (the notion of limit along an ultrafilter, the Stone-Čech compactification βX of a discrete space X, etc.), and in Banach spaces (the so-called ultraproduct technique).In 1975, F. Galvin and S. Glazer found a beautiful ultrafilter proof of Hindman's theorem, namely the property that for every finite partition of the natural numbers N = C 1 ∪ . . . ∪ C r , there exists an infinite set X and a piece C i such that all sums of distinct elements from X belong to C i . Since this time, ultrafilters on N have been successfully used also in combinatorial number theory and in Ramsey theory. The key fact is that the compact space βN of ultrafilters on N can be equipped 2000 Mathematics Subject Classification. 03H05, 03E05, 54D80.

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Section: Hyper-shiftsmentioning

confidence: 99%

Section: Final Remarks and Open Questionsmentioning

confidence: 99%

Hypernatural Numbers as Ultrafilters

Nasso

2015

Nonstandard Analysis for the Working Mathematician

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“…Both in Blass and Di Nasso [1] and in Luperi Baglini [7] it has been shown that the relation of finite embeddability between sets has a very nice characterization in terms of nonstandard analysis, which allows to study some of its properties in a quite simple and elegant way. We recall the characterization (in the following proposition, it is assumed 3 An equation P(x 1 , ..., x n ) = 0 is partition regular if and only if for every finite coloration N = C 1 ∪ ... ∪ C n of N there exists an index i and monochromatic elements a 1 , ..., a n ∈ C i such that P(a 1 , ..., a n ) = 0.…”

Section: A Nonstandard Proof Of the Main Resultsmentioning

confidence: 99%

“…For completeness, even if we will not use this fact, we also recall that Corollary 3.7 can be improved: in fact (as proved by Blass and Di Nasso in [1] and by Luperi Baglini in [7]) for every U, V ∈ βN we have U, V ≤ fe U ⊕ V.…”

Section: Luperi Baglinimentioning

confidence: 99%

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10.4115/jla.v6i0.226

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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-Čech compactification of the discrete space of natural 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.

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On Hausdorff operators in ZF$\mathsf {ZF}$

Keremedis

Tachtsis

2023

Mathematical Logic Qtrly

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A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with , , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff spaces are effectively Hausdorff” in . The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in . The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of . The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”.

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Finite Embeddability of Sets and Ultrafilters

Cited by 9 publications

References 4 publications

Hypernatural Numbers as Ultrafilters

Hypernatural Numbers as Ultrafilters

10.4115/jla.v6i0.226

On Hausdorff operators in ZF$\mathsf {ZF}$

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