On finite approximations of topological algebraic systems
| dc.creator | Glebsky, L. Yu. | |
| dc.creator | Gordon, E. I. | |
| dc.creator | Henson, C. W. | |
| dc.date | 2003-11-21 | |
| dc.date | 2006-07-31 | |
| dc.date.accessioned | 2026-07-07T06:35:49Z | |
| dc.date.available | 2026-07-07T06:35:49Z | |
| dc.description | We introduce and discuss a definition of approximation of a topological algebraic system $A$ by finite algebraic systems of some class $\K$. For the case of a discrete algebraic system this definition is equivalent to the well-known definition of a local embedding of an algebraic system $A$ in a class $\K$ of algebraic systems. According to this definition $A$ is locally embedded in $K$ iff it is a subsystem of an ultraproduct of some systems in $\K$. We obtain a similar characterization of approximation of a locally compact system $A$ by systems in $\K$. We inroduce the bounded formulas of the signature of $A$ and their approximations similar to those introduced by C.W.Henson \cite{he} for Banach spaces. We prove that a positive bounded formula $\f$ holds in $A$ if all precise enough approximations of $\f$ hold in all precise enough approximations of $A$. We prove that a locally compact field cannot be approximated by finite associative rings (not necessary commutative). Finite approximations of the field $\R$ can be concedered as computer systems for reals. Thus, it is impossible to construct a computer arithmetic for reals that is an associative ring. | |
| dc.description | 20 pages, sent to Journal of Symbolic Logic | |
| dc.identifier | https://arxiv.org/abs/math/0311387 | |
| dc.identifier | http://arxiv.org/abs/math/0311387 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/99908 | |
| dc.subject | Logic | |
| dc.subject | Rings and Algebras | |
| dc.subject | 26E35, 03H05; Secondary 28E05, 42A38 | |
| dc.title | On finite approximations of topological algebraic systems | |
| dc.type | text |