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.