Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2025-01-03T18:54:32.952Z Has data issue: false hasContentIssue false
  • English
  • Français

Computability and the algebra of fields: Some affine constructions

Published online by Cambridge University Press:  12 March 2014

J. V. Tucker
J. V. Tucker*
Affiliation:
Universitetet i Oslo, Blindern, Oslo 3, Norway
*
Mathematisch Centrum, Tweede Boerhaavestraat 49, 1091A1 Amsterdam, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Extract

A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure A = (A; σ1,…, σk) is computable if it possesses a recursive coordinate system in the following precise sense: associated to A there is a pair (α, Ω) consisting of a recursive set of natural numbers Ω and a surjection α: Ω → A so that (i) the relation defined on Ω by n ≡α m iff α(n) = α(m) in A is recursive, and (ii) each of the operations of A may be effectively followed in Ω, that is, for each (say) r-ary operation σ on A there is an r argument recursive function on Ω which commutes the diagram

wherein αr is r-fold α × … × α.

This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra—an isomorphism invariant possessed of all finite algebraic systems—and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 1 , March 1980 , pp. 103 - 120
Copyright
Copyright © Association for Symbolic Logic 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arnold, V.I., Ordinary differential equations (translated by Silverman, R.A.), M.I.T. Press, Cambridge, Massachusetts, 1973.Google Scholar
[2]Bellman, R. and Kalaba, R. (Editors), Selected papers on mathematical trends in control theory, Dover, New York, 1964.Google Scholar
[3]Browder, F.E. (Editor), Mathematical developments arising from Hubert's problems, American Mathematical Society, Providence, Rhode Island, 1976.Google Scholar
[4]Cohn, P.M., Universal algebra, Harper & Row, New York, 1965.Google Scholar
[5]Duffin, R.J., Algorithms for classical stability problems, SIAM Review, vol. 11 (1969), pp. 196–213.CrossRefGoogle Scholar
[6]Ershov, Y.L., Numbered fields, Logic, methodology and philosophy of science. III (van Rootselaar, B. and Staal, J.F., Editors), North-Holland, Amsterdam, 1968, pp. 31–34.CrossRefGoogle Scholar
[7]Fröhlich, A. and Shepherdson, J.C., Effective procedures in field theory, Philosophical Transactions of the Royal Society of London (A), vol. 248 (1956), pp. 407–432.Google Scholar
[8]Humphreys, J.E., Linear algebraic groups, Springer-Verlag, New York, 1975.CrossRefGoogle Scholar
[9]Kreisel, G. and Krivine, J.L., Elements of mathematical logic, North-Holland, Amsterdam, 1971.Google Scholar
[10]Lachlan, A. H. and Madison, E. W., Computable fields and arithmetically definable ordered fields, Proceedings of the American Mathematical Society, vol. 24 (1970), pp. 803–807.CrossRefGoogle Scholar
[11]Levine, H.I., Singularities of differentiable mappings, Proceedings of the Liverpool Singularities Symposium 1 (Wall, C.T.C., Editor), Springer-Verlag, Berlin, 1971, pp. 1–89.Google Scholar
[12]Madison, E. W., A note on computable real fields, this Journal, vol. 35 (1970), pp. 239–241.Google Scholar
[13]Madison, E. W., Some remarks on computable (non-Archimedean) ordered fields, Journal of the London Mathematical Society, vol. 4 (1971), pp. 304–308.CrossRefGoogle Scholar
[14]Mal'cev, A. I., Constructive algebras. I, The meta-mathematics of algebraic systems. Collected papers: 1967–1976 (translated and edited by Wells, B.F. III), North-Holland, Amsterdam, 1971, pp. 148–214.Google Scholar
[15]Mal'cev, A. I., Algebraic systems (translated by Seckler, B.D. and Doohovskoy, A. P.), Springer-Verlag, Berlin, 1973.CrossRefGoogle Scholar
[16]Miller, C. F. III, On group-theoretic decision problems and their ctassification, Princeton University Press, Princeton, New Jersey, 1971.Google Scholar
[17]Moschovakis, Y. N., Notation systems and recursive ordered fields, Compositio Mathematica, vol. 17 (1965), pp. 40–71.Google Scholar
[18]Rabin, M. O., Computable algebra, general theory and the theory of computable fields, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–360.Google Scholar
[19]Rice, H. G., Recursive real numbers, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 784–791.CrossRefGoogle Scholar
[20]Seidenberg, A., A new decision method for elementary algebra, Annals of Mathematics, vol. 60 (1954), pp. 365–374.CrossRefGoogle Scholar
[21]Schafer, R. D., An introduction to non-associative algebras, Academic Press, New York, 1966.Google Scholar
[22]Shafarevich, I. R., Basic algebraic geometry, Springer-Verlag, Berlin, 1974.CrossRefGoogle Scholar
[23]Tarski, A., A decision method for elementary algebra and geometry, University of California Press, Berkeley, 1951.CrossRefGoogle Scholar
[24]Tarski, A., What is elementary geometry?The philosophy of mathematics (Hintikka, J., Editor), Oxford University Press, London, 1969, pp. 164–175.Google Scholar
[25]Tucker, J.V., Computability as an algebraic property. Part one: general theory (in preparation).Google Scholar
[26]Tucker, J.V., Computability as an algebraic property. Part two: applications (in preparation).Google Scholar
[27]Waerden, B. L. van der, Modern algebra. I (translated by Blum, F.), Ungar, New York, 1949.Google Scholar
[28]Waerden, B. L. van der, Algebra. I (translated by Blum, F. and Schulenberger, J. R.), Ungar, New York, 1970.Google Scholar