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A notion of effectiveness in arbitrary structures

Published online by Cambridge University Press:  12 March 2014

W. M. Lambert Jr
W. M. Lambert Jr*
Affiliation:
Loyola University of Los Angeles
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Extract

The well-developed theory of recursion has been applied to certain algebraic structures with countable universes to obtain interesting and natural results on the question of effective procedures in such structures ([2], [9], [10], [13]). It has also been applied to structures related to the real numbers, via approximations; we cite [3] as but one example. We are interested, however, in the problem of what operations in an arbitrary algebraic structure, possibly quite large and unnatural, can reasonably be considered “effective” in terms of the underlying operations and relations of the structure.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 4 , January 1969 , pp. 577 - 602
Copyright
Copyright © Association for Symbolic Logic 1969

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References

[1]Davis, M., Computability and unsolvability, McGraw-Hill, New York, 1958.Google Scholar
[2]Froehlich, A. and Shepherdson, J., Effective procedures in field theory, Philosophical Transactions of the Royal Society of London, vol. 248 (1956), pp. 407–432.Google Scholar
[3]Goodstein, R., Recursive analysis, North-Holland, Amsterdam, 1961.Google Scholar
[4]Hermann, G., Die Frage der endliche viele Schritte in der Theorie der Polynomideale, Mathematische Annalen, vol. 95 (1926), pp. 736–788.CrossRefGoogle Scholar
[5]Jacobson, N., Lectures in abstract algebra, Vol. 2, Princeton Univ. Press, Princeton, N.J., 1952.Google Scholar
[6]Kleene, S., Introduction to metamathematics, Princeton Univ. Press, Princeton, 1952.Google Scholar
[7]Lambert, W., Effectiveness, elementary definability and prime polynomial ideals, dissertation, U.C.L.A., 1965.Google Scholar
[8]Lambert, W., Elementary characterization of “prime polynomial ideal” (Abstract, Notices of the American Mathematical Society, vol. 12 (1965), p. 619.Google Scholar
[9]Mal′cev, A., Constructive algebras I, Russian Mathematical Surveys, vol. 16 (1961), pp. 77–129.Google Scholar
[10]Mal′cev, A., Recursive abellan groups, Soviet Mathematics Doklady, vol. 3 (1962), pp. 1431–1433.Google Scholar
[11]Noether, E., Ein algebraisches Kriterium für absolute Irreduzibilität, Mathematische Annalen, vol. 85 (1922), pp. 26–33.CrossRefGoogle Scholar
[12]Noether, E., Eliminationstheorie und allgemeine Idealtheorie, Mathematische Annalen, vol. 90 (1923), pp. 229–261.CrossRefGoogle Scholar
[13]Rabin, M., Computable algebra, general theory and theory of computable fields, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–360.Google Scholar
[14]Robinson, A., Introduction to model theory and to the metamathematics of algebra, North-Holland, Amsterdam, 1963.Google Scholar
[15]Robinson, A., Les rapports entre le calcul déductif et l'interpretation sémantique d'une système axiomatique, Colloque Internationale: Les méthodes formelles en axiomatique, Paris, pp. 35–52 (1950, published 1953).Google Scholar
[16]van der Waerden, B., Eine Bemerkungen über die Unzerlegbarkeit von Polynomen, Mathematische Annalen, vol. 102 (1930), pp. 738–739.CrossRefGoogle Scholar
[17]van der Waerden, B., Moderne Algebra, 2nd ed., Springer, Berlin, 1937.CrossRefGoogle Scholar