Model companion of ordered theories with an automorphism. (English) Zbl 1347.03065
The paper studies the question of the existence of a model companion of a theory expanded by an automorphism.
The authors obtain a complete characterization of all model-complete theories expanding the theory of non-trivial dense linear orders without endpoints in the language \(\{<,\sigma\}\) where \(\sigma\) is an increasing automorphism.
They also obtain a complete characterization of all model-complete theories of omega-sums in the class of ordered abelian groups in the language \(\{+, -, 0, <,\sigma\}\) where \(\sigma\) is a “piecewise” increasing automorphism.
At last, they show that the theory of real closed fields does not have a model companion in the language \(\{+, -, \times, 0, 1, <, \sigma\}\) where \(\sigma\) is an increasing automorphism.
Reviewer: Beibut Kulpeshov (Almaty)
MSC:
03C10 | Quantifier elimination, model completeness, and related topics |
03C64 | Model theory of ordered structures; o-minimality |
20A05 | Axiomatics and elementary properties of groups |
20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
Keywords:
model-complete theory; automorphism; expansion of theory; ordered theory; ordered abelian group; real closed field; o-minimalityReferences:
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