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

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.