ON FINITE APPROXIMATIONS OF TOPOLOGICAL ALGEBRAIC SYSTEMS.
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 s...
| Publicado en: | Journal of Symbolic Logic Vol. 72; no. 1; pp. 1 - 26 |
|---|---|
| Autores principales: | , , |
| Formato: | Artículo |
| Publicado: |
Cambridge University Press
Mar2007
|
| Materias: | |
| Acceso en línea: | Ver este registro en EBSCOhost |
| fields | @attributes: recordID: 1 pdfLink: plink: https://search.ebscohost.com/login.aspx?direct=true&db=hlh&AN=24665154&site=ehost-live header: @attributes: shortDbName: hlh uiTerm: 24665154 longDbName: Humanities International Complete uiTag: AN controlInfo: bkinfo: jinfo: jid: 00224812 3TY jtl: Journal of Symbolic Logic issn: 00224812 maglogo: N pubinfo: dt: Mar2007 vid: 72 iid: 1 pid: 15979 pub: Cambridge University Press artinfo: ui: 24665154 10.2178/jsl/1174668381 ppf: 1 ppct: 25 formats: tig: atl: ON FINITE APPROXIMATIONS OF TOPOLOGICAL ALGEBRAIC SYSTEMS. aug: au: Glebsky, L. Yu Gordon, E. I. Henson, C. Ward affil: Universidad Autonoma De San Luis Potosi, Instituto de Investigacion En Communicacion Optica, Avrakakorum 1470, Lomas 4ta Session, San Luis Potosi SLP 7820, Mexico Department of Mathematics and Computer Science, Eastern Illinois University, 600 Lincoln Avenue, Charleston, IL 61920-3099, USA Department of Mathematics, University of Illinois, Urban-Champaign, 1409 West Green Street, Urbana, IL 61801, USA su: Mathematical logic Topological algebras Algebraic topology Banach spaces Topology Ring theory Computer arithmetic Associative rings sug: subj: Mathematical logic Topological algebras Algebraic topology Banach spaces Topology Ring theory Computer arithmetic Associative rings ab: 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. pubtype: Academic Journal doctype: Article src: R language: English refInfo: copyright: @attributes: flag: Y dt: @attributes: year: 2007 holdings: @attributes: islocal: N |
|---|