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Transactions of the American Mathematical Society

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Nonnegative solvability of linear equations in certain ordered rings

Author: Philip Scowcroft
Journal: Trans. Amer. Math. Soc. 358 (2006), 3535-3570
MSC (2000): Primary 03C64; Secondary 06F20, 15A39
Published electronically: March 1, 2006
MathSciNet review: 2218988
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Abstract: In the integers and in certain densely ordered rings that are not fields, projections of the solution set of finitely many homogeneous weak linear inequalities may be defined by finitely many congruence inequalities, where a congruence inequality combines a weak inequality with a system of congruences. These results extend well-known facts about systems of weak linear inequalities over ordered fields and imply corresponding analogues of Farkas' Lemma on nonnegative solvability of systems of linear equations.

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  • 1. C. C. Chang and H. J. Keisler, Model theory, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Studies in Logic and the Foundations of Mathematics, Vol. 73. MR 0409165
  • 2. George B. Dantzig, Linear programming and extensions, Reprint of the 1968 corrected edition, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1998. MR 1658673
  • 3. L. van den Dries, Some applications of a model theoretic fact to (semi-)algebraic geometry, Nederl. Akad. Wetensch. Indag. Math. $ \mathbf{44}$ (1982), 397-401. MR 0683527 (84m:14029)
  • 4. J. Farkas, Theorie der einfachen Ungleichungen, J. Reine Angew. Math. $ \mathbf{124}$ (1902), 1-27.
  • 5. Robert W. Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR 0229624
  • 6. J. H. Grace and A. Young, The algebra of invariants, University Press, Cambridge, 1903.
  • 7. P. Rothmaler, Some model theory of modules. II. On stability and categoricity of flat modules, J. Symbolic Logic $ \mathbf{48}$ (1983), 970-985. MR 0727787 (85e:03084)
  • 8. V. Weispfenning, Elimination of quantifiers for certain ordered and lattice-ordered abelian groups, Bull. Soc. Math. Belg. Sér. B $ \mathbf{33}$ (1981), 131-155. MR 0620968 (82h:03022)
  • 9. Volker Weispfenning, Model theory of abelian 𝑙-groups, Lattice-ordered groups, Math. Appl., vol. 48, Kluwer Acad. Publ., Dordrecht, 1989, pp. 41–79. MR 1036073
  • 10. V. Weispfenning, Existential equivalence of ordered abelian groups with parameters, Arch. Math. Logic 29 (1990), no. 4, 237–248. MR 1062728,
  • 11. H. Weyl, The elementary theory of convex polyhedra, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N.J., 1950, pp. 3–18. MR 0038088

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Additional Information

Philip Scowcroft
Affiliation: Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459

Received by editor(s): July 19, 2004
Published electronically: March 1, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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