Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T15:41:42.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Functorial properties of algebraic closure and Skolemization

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

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Crossley, J. N. and Nerode, A. (1974), Combinatorial functors (Ergebnisse der Math. und ihrer Grenzebiete 81, Springer-Verlag, Berlin).CrossRefGoogle Scholar
Hodges, W. (1974), ‘A normal form for algebraic constructions’, Bull. London Math. Soc. 6, 57–60.CrossRefGoogle Scholar
Jacobson, N. (1964), Lectures in abstract algebra, Volume III (Van Nostrand, New York).CrossRefGoogle Scholar
Keisler, H. J. (1976), ‘Six classes of theories’, J. Austral. Math. Soc. Ser. A 21, 257–266.CrossRefGoogle Scholar
Morley, H. D. (1965), ‘Categoricity in power’, Trans. Amer. Math. Soc. 114, 514–538.CrossRefGoogle Scholar
Shelah, S. (1971), ‘Stability, the f.c.p. and superstability’, Ann. Math. Logic 3, 271–362.CrossRefGoogle Scholar