Ordered fields and $${{{\rm \L}\Pi\frac{1}{2}}}$$-algebras

  • Authors:

  • Enrico Marchioni

  • Affiliations:

  • Open University of Catalonia, Department of Information and Communication Sciences, Rambla del Poblenou 156, 08018, Barcelona, Spain

  • Venue:
  • Soft Computing - A Fusion of Foundations, Methodologies and Applications
  • Year:
  • 2009

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Abstract

In this work we further explore the connection between $${{{\rm \L}\Pi\frac{1}{2}}}$$-algebras and ordered fields. We show that any two $${{{\rm \L}\Pi\frac{1}{2}}}$$-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 $${{{\rm \L}\Pi\frac{1}{2}}}$$-chain generates the whole variety if and only if it contains a subalgebra isomorphic to the $${{{\rm \L}\Pi\frac{1}{2}}}$$-chain of real algebraic numbers, that consequently is the smallest $${{{\rm \L}\Pi\frac{1}{2}}}$$-chain generating the whole variety. We also show that any two different subalgebras of the $${{{\rm \L}\Pi\frac{1}{2}}}$$-chain over the real algebraic numbers generate different varieties. This will be exploited in order to prove that the lattice of subvarieties of $${{{\rm \L}\Pi\frac{1}{2}}}$$-algebras has the cardinality of the continuum. Finally, we will also briefly deal with some model-theoretic properties of $${{{\rm \L}\Pi\frac{1}{2}}}$$-chains related to real closed fields, proving quantifier-elimination and related results.