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Functorial properties of algebraic closure and Skolemization

Part of: Computability and recursion theory General theory of categories and functors Model theory

Published online by Cambridge University Press:  09 April 2009

C. J. Ash  and
A. Nerode
C. J. Ash
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
A. Nerode
Affiliation:
Department of Mathematics, Cornell University, Ithaca, N. Y. 14853, U.S.A.
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Abstract

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It is shown that no functor F exists from the category of sets with injections, to the category of algebraically closed fields of given characteristic, with monomorphisms, having the properties that for all sets A. F(A) is an algebraically closed field having transcendence base A and for all injections f. F(f) extends f. There does exist such a functor from the category of linearly-ordered sets with order monomorphisms.

An application to model-theory using the same methods is given showing that while the theory of algebraically closed fields is ω-stable, its Skolemization is not stable in any power.

MSC classification

Secondary: 18A15: Foundations, relations to logic and deductive systems 03C45: Classification theory, stability and related concepts 03C60: Model-theoretic algebra 03D50: Recursive equivalence types of sets and structures, isols
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 2 , August 1981 , pp. 136 - 141
Copyright
Copyright © Australian Mathematical Society 1982

References

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