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The elementary class of products of totally ordered abelian group

Published online by Cambridge University Press:  12 March 2014

Daniel Gluschankof
Daniel Gluschankof*
Affiliation:
Département de Mathématique, Université d'Angers, 49000 Angers, France
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Extract

A basic goal in model-theoretic algebra is to obtain the classification of the complete extensions of a given (first-order) algebraic theory.

Results of this type, for the theory of totally ordered abelian groups, were obtained first by A. Robinson and E. Zakon [5] in 1960, later extended by Yu. Gurevich [4] in 1964, and further clarified by P. Schmitt in [6].

Within this circle of ideas, we give in this paper an axiomatization of the first-order theory of the class of all direct products of totally ordered abelian groups, construed as lattice-ordered groups (l-groups)—see the theorem below. We think of this result as constituing a first step—undoubtedly only a small one—towards the more general goal of classifying the first-order theory of abelian l-groups.

We write groups for abelian l-groups construed as structures in the language 〈 ∨, ∧, +, −, 0〉 (“−” is an unary operation). For unproved statements and unexplicated definitions, the reader is referred to [1].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 1 , March 1991 , pp. 295 - 299
Copyright
Copyright © Association for Symbolic Logic 1991

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References

REFERENCES

[1]Bigard, A., Keimel, K. and Wolfenstein, S., Groupes et anneaux réticulées, Lecture Notes in Mathematics, vol. 608, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
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[5]Robinson, A. and Zakon, E., Elementary properties of ordered abelian groups, Transactions of the American Mathematical Society, vol. 96 (1960), pp. 222–236.CrossRefGoogle Scholar
[6]Schmitt, P. H., Model- and substructure-complete theories of ordered abelian groups, Models and sets (proceedings of Logic Colloquium '83), Lecture Notes in Mathematics, vol. 1103, Springer-Verlag, Berlin, 1984, pp. 389–418.Google Scholar