Ordered fields and {{{\rm \L}\Pi\frac{1}{2}}} -algebras |
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Authors: | Enrico Marchioni |
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Affiliation: | (1) Department of Information and Communication Sciences, Open University of Catalonia, Rambla del Poblenou 156, 08018 Barcelona, Spain |
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Abstract: | In this work we further explore the connection between -algebras and ordered fields. We show that any two -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 -chain generates the whole variety if and only if it contains a subalgebra isomorphic to the -chain of real algebraic numbers, that consequently is the smallest -chain generating the whole variety. We also show that any two different subalgebras of the -chain over the real algebraic numbers generate different varieties. This will be exploited in order to prove that the lattice
of subvarieties of -algebras has the cardinality of the continuum. Finally, we will also briefly deal with some model-theoretic properties of
-chains related to real closed fields, proving quantifier-elimination and related results. |
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Keywords: | -algebras" target="_blank">gif" alt="$${{{\rm \L}\Pi\frac{1}{2}}}$$" align="middle" border="0"> -algebras Ordered fields Real closed fields |
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