Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 507-533

ON MODEL THEORY OF COVERS OF ALGEBRAICALLY CLOSED FIELDS

Tapani Hyttinen and Kaisa Kangas

University of Helsinki, Department of Mathematics and Statistics
P.O. Box 68, FI-00014 University of Helsinki, Finland; tapani.hyttinen 'at' helsinki.fi

University of Helsinki, Department of Mathematics and Statistics
P.O. Box 68, FI-00014 University of Helsinki, Finland; kaisa.kangas 'at' helsinki.fi

Abstract. We study covers of the multiplicative group of an algebraically closed field as quasiminimal pregeometry structures and prove that they satisfy the axioms for Zariski-like structures presented in [7], Section 4. These axioms are intended to generalize the concept of a Zariski geometry into a non-elementary context. In the axiomatization, it is required that for a structure M, there is, for each n, a collection of subsets of Mⁿ, that we call the irreducible sets, satisfying certain properties. These conditions are generalizations of some qualities of irreducible closed sets in the Zariski geometry context. They state that some basic properties of closed sets (in the Zariski geometry context) are satisfied and that specializations behave nicely enough. They also ensure that there are some traces of Compactness even though we are working in a non-elementary context.

2010 Mathematics Subject Classification: Primary 12L12, 03C60.

Key words: Model-theoretic algebra, covers of algebraically closed fields.

Reference to this article: T. Hyttinen and K. Kangas: On model theory of covers of algebraically closed fields. Ann. Acad. Sci. Fenn. Math. 40 (2015), 507-533.

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doi:10.5186/aasfm.2015.4048