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Hyper-regular lattice-ordered groups

Published online by Cambridge University Press:  12 March 2014

Daniel Gluschankof  and
François Lucas
Daniel Gluschankof
Affiliation:
Equipe de Logique Mathématique, UFR de Mathématiques, Université Paris 7, CNRS (URA 753) 75251 Paris Cedex 05, France, E-mail gluscha@logique.jussieu.fr, grpord@logique.jussieu.fr
François Lucas
Affiliation:
Département de Mathématiques, Université d'Angers, 49045 Angers Cedex, France, E-mail: lucas@univ-rennesl.fr
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Extract

It is a well-known fact that the notion of an archimedean order cannot be formalized in the first-order calculus. In [12] and [18], A. Robinson and E. Zakon characterized the elementary class generated by all the archimedean, totally-ordered abelian groups (o-groups) in the language 〈+,<〉, calling it the class of regularly ordered or generalized archimedean abelian groups. Since difference (−) and 0 are definable in that language, it is immediate that in the expanded language 〈 +, −, 0, < 〉 the definable expansion of the class of regular groups is also the elementary class generated by the archimedean ones. In the more general context of lattice-ordered groups (l-groups), the notion of being archimedean splits into two different notions: a strong one (being hyperarchimedean) and a weak one (being archimedean). Using the representation theorem of K. Keimel for hyperarchimedean l-groups, we extend in this paper the Robinson and Zakon characterization to the elementary class generated by the prime-projectable, hyperarchimedean l-groups. This characterization is also extended here to the elementary class generated by the prime-projectable and projectable archimedean l-groups (including all complete l-groups). Finally, transferring a result of A. Touraille on the model theory of Boolean algebras with distinguished ideals, we give the classification up to elementary equivalence of the characterized class.

We recall that a lattice-ordered group, l-group for short, is a structure

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 4 , December 1993 , pp. 1342 - 1358
Copyright
Copyright © Association for Symbolic Logic 1993

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References

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