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The laws of integer divisibility, and solution sets of linear divisibility conditions

  • L. van den Dries (a1) and A. J. Wilkie (a2)

Abstract

We prove linear and polynomial growth properties of sets and functions that are existentially definable in the ordered group of integers with divisibility. We determine the laws of addition with order and divisibility.

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[1]Bel'tyukov, A. P.. Decidability of the universal theory of natural numbers with + and ∣, Seminars of Steklov Mathematical Institute (Leningrad), vol. 60 (1976), pp. 1528.
[2]van den Dries, L., Quantifier elimination for linear formulas over ordered and valued fields. Bulletin de la Société Mathématique de Belgique, vol. 33 (1981). pp. 1933.
[3]Groemer, H., On the extension of additive functionals on classes of convex sets, Pacific Journal of Mathematics, vol. 75 (1978), pp. 397410.
[4]Lipshitz, L., The diophantine problem for addition and divisibility, Transactions of the American Mathematical Society, vol. 235 (1978), pp. 271283.
[5]Lipshitz, L., Some remarks on the diophantine problem for addition and divisibility. Bulletin de la Société Mathématique de Belgique, vol. 33 (1981), pp. 4152.
[6]Macintyre, A., A theorem of Rabin in a general setting. Bulletin de la Société Mathématique de Belgique, vol. 33 (1981), pp. 5363.
[7]Moschovakis, Y., On primitive recursive algorithms and the greatest common divisor function, Theoretical Computer Science, to appear.
[8]Wilkie, A.J., Modèles non standard de l'arithmétique, et complexité algorithmique. Modèles non standard en arithmétique et théorie des ensembles. Publications Mathématiques de l'Université Paris VII, 1984, pp. 545.

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