The Puritz Order and Its Relationship to the Rudin-Keisler Order

“…In those papers, as well as in Puritz' follow-up [44], new results about the Rudin-Keisler ordering were proved by nonstandard methods, along with new characterizations of special ultrafilters, such as P-points and selective ultrafilters. (See also [42], where the study of similar properties as in Puritz' papers was continued.) In [7], A. Blass pushed that approach further and provided a comprehensive treatment of ultrafilter properties as reflected by the nonstandard numbers of the associated ultrapowers.…”

Section: ])mentioning

confidence: 89%

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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“…The adjective "directed" refers to the so called Puritz order, a preordering of nonstandard models corresponding to the Rudin-Keisler (pre)ordering of ultrafilters (see [12,11]):…”

Section: Functional Extensionsmentioning

confidence: 99%

A simple algebraic characterization of nonstandard extensions

Forti

2012

Proc. Amer. Math. Soc.

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We introduce the notion of functional extension of a set X, by means of two natural algebraic properties of the operator " * " on unary functions. We study the connections with ultrapowers of structures with universe X, and we give a simple characterization of those functional extensions that correspond to limit ultrapower extensions. In particular we obtain a purely algebraic proof of Keisler's characterization of nonstandard (= complete elementary) extensions.

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The Puritz Order and Its Relationship to the Rudin-Keisler Order

Cited by 17 publications

References 10 publications

The eightfold path to nonstandard analysis

The eightfold path to nonstandard analysis

Hypernatural Numbers as Ultrafilters

A simple algebraic characterization of nonstandard extensions

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