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Ordered fields and ŁΠ1/2-algebras

Ordered fields and ŁΠ1/2-algebras
Ordered fields and ŁΠ1/2-algebras
In this work we further explore the connection between ŁΠ1/ 2-algebras and ordered fields. We show that any two ŁΠ1/ 2-chains generate the same variety if and only if they are related to ordered fields that have the same universal theory. This will yield that any ŁΠ1/2-chain generates the whole variety if and only if it contains a subalgebra isomorphic to the ŁΠ1/2-chain of real algebraic numbers, that consequently is the smallest ŁΠ1/ 2-chain generating the whole variety. We also show that any two different subalgebras of the ŁΠ1/2-chain over the real algebraic numbers generate different varieties. This will be exploited in order to prove that the lattice of subvarieties of ŁΠ1/ 2-algebras has the cardinality of the continuum. Finally, we will also briefly deal with some model-theoretic properties of ŁΠ1/ 2-chains related to real closed fields, proving quantifier-elimination and related results. © Springer-Verlag 2008.
????1/2, Ordered fields, Real closed fields
559-564
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96

Marchioni, Enrico (2009) Ordered fields and ŁΠ1/2-algebras. Soft Computing, 13 (6), 559-564. (doi:10.1007/s00500-008-0315-y).

Record type: Article

Abstract

In this work we further explore the connection between ŁΠ1/ 2-algebras and ordered fields. We show that any two ŁΠ1/ 2-chains generate the same variety if and only if they are related to ordered fields that have the same universal theory. This will yield that any ŁΠ1/2-chain generates the whole variety if and only if it contains a subalgebra isomorphic to the ŁΠ1/2-chain of real algebraic numbers, that consequently is the smallest ŁΠ1/ 2-chain generating the whole variety. We also show that any two different subalgebras of the ŁΠ1/2-chain over the real algebraic numbers generate different varieties. This will be exploited in order to prove that the lattice of subvarieties of ŁΠ1/ 2-algebras has the cardinality of the continuum. Finally, we will also briefly deal with some model-theoretic properties of ŁΠ1/ 2-chains related to real closed fields, proving quantifier-elimination and related results. © Springer-Verlag 2008.

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More information

e-pub ahead of print date: 21 May 2008
Published date: April 2009
Keywords: ????1/2, Ordered fields, Real closed fields

Identifiers

Local EPrints ID: 416726
URI: http://eprints.soton.ac.uk/id/eprint/416726
PURE UUID: a216edaf-83e3-487c-85a9-5af3a97134dd

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Date deposited: 05 Jan 2018 17:30
Last modified: 13 Mar 2019 19:04

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Author: Enrico Marchioni

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