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On finite approximations of topological algebraic systems

Published online by Cambridge University Press:  12 March 2014

L. Yu. Glebsky ,
E. I. Gordon  and
C. Ward Henson
L. Yu. Glebsky
Affiliation:
Universidad Autonoma de San Luis Potosi, Instituto de Investigacion en Communicacion Optica, Avkarakorum 1470, Lomas 4TA Session, San Luis Potosi SLP 7820, Mexico. E-mail: glebsky@cactus.iico.uaslp.mx
E. I. Gordon
Affiliation:
Eastern Illinois University, Department of Mathematics and Computer Science, 600 Lincoln Avenue, Charleston, IL 61920-3099, USA. E-mail: yigordon@eiu.edu
C. Ward Henson
Affiliation:
University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 West Green Street, Urbana, IL 61801, USA. E-mail: henson@math.uiuc.edu URL: http://www.math.uiuc.edu/˜henson
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Abstract

We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class . If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class of algebraic systems. One characterization of this concept states that A is locally embedded in iff it is a subsystem of an ultraproduct of systems from . In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 1 - 25
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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