Hyperfinite Transversal Theory

Živaljević, Boško

doi:10.2307/2154170

Cited by 3 publications

(3 citation statements)

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“…Remark 1 Nonstandard analysis versions of P.Hall's theorem were investigated in [13] Lemma 1 (3) and Theorem 4 imply that the set F may be equipped with operation ⊙, satisfying definition of l-quasigroup. Indeed, by the condition 3) the system {A g,h | h ∈ F } satisfies Theorem 4 for any fixed g ∈ F and thus for any g, h ∈ F there exists g ⊙ h ∈ A g,h such that for any g ∈ F {g ⊙ h | h ∈ F } is a permutation of F .…”

Section: Proof Of Theoremmentioning

confidence: 99%

On finite approximations of topological algebraic systems

Glebsky¹,

Gordon²,

Hensen³

2007

J. symb. log.

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We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from K using the language of nonstandard analysis.In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15,16]). We prove that a positive bounded formula ϕ holds in A if and only if all precise enough approximations of ϕ hold in all precise enough approximations of A.We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field R can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings.

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Section: Proof Of Theoremmentioning

confidence: 99%

On finite approximations of topological algebraic systems

Glebsky¹,

Gordon²,

Hensen³

2007

J. symb. log.

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“…Most of the results in this paper were proved during the March 1987, Oberwolfach meeting on nonstandard analysis, and the authors are grateful for the opportunity that this meeting provided for them to work together. Subsequently the methods have been refined and generalized by Zivaljevic in his thesis [16] and in a series of papers [17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Analytic mappings on hyperfinite sets

Henson¹,

Ross²

1993

Proc. Amer. Math. Soc.

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Let S and T be hyperfinite sets in an N]-saturated nonstandard universe. The following are equivalent: m 1*1 ~ l (l) rff ~ • • (ii) There is a bijection from 5 onto T whose graph is Borel (over the internal subsets of S x T). This follows from somewhat more general results about analytic partial functions on hyperfinite sets, the proofs of which use Choquet's theorem on the capacitibility of analytic sets. This paper includes: proofs of the above results; an elementary direct construction for extensions of internal set functions to capacities; and a surprising corollary asserting the nonexistence of ergodic Borel transformations of a Loeb probability space.

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“…Subsequently the methods have been refined and generalized by Zivaljevic in his thesis [16] and in a series of papers [17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Analytic Mappings on Hyperfinite Sets

Henson¹,

Ross²

1993

Proceedings of the American Mathematical Society

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Abstract.Let S and T be hyperfinite sets in an N]-saturated nonstandard universe. The following are equivalent:There is a bijection from 5 onto T whose graph is Borel (over the internal subsets of S x T). This follows from somewhat more general results about analytic partial functions on hyperfinite sets, the proofs of which use Choquet's theorem on the capacitibility of analytic sets. This paper includes: proofs of the above results; an elementary direct construction for extensions of internal set functions to capacities; and a surprising corollary asserting the nonexistence of ergodic Borel transformations of a Loeb probability space.

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Hyperfinite Transversal Theory

Cited by 3 publications

References 0 publications

On finite approximations of topological algebraic systems

On finite approximations of topological algebraic systems

Analytic mappings on hyperfinite sets

Analytic Mappings on Hyperfinite Sets

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