Nonnegative solvability of linear equations in certain ordered rings
Author:
Philip Scowcroft
Journal:
Trans. Amer. Math. Soc. 358 (2006), 35353570
MSC (2000):
Primary 03C64; Secondary 06F20, 15A39
Published electronically:
March 1, 2006
MathSciNet review:
2218988
Fulltext PDF Free Access
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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 wellknown 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.
 1.
C.
C. Chang and H.
J. Keisler, Model theory, NorthHolland Publishing Co.,
AmsterdamLondon; American Elsevier Publishing Co., Inc., New York, 1973.
Studies in Logic and the Foundations of Mathematics, Vol. 73. MR 0409165
(53 #12927)
 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
(99g:90004)
 3.
Lou
van den Dries, Some applications of a model theoretic fact to
(semi) algebraic geometry, Nederl. Akad. Wetensch. Indag. Math.
44 (1982), no. 4, 397–401. MR 683527
(84m:14029)
 4.
J. Farkas, Theorie der einfachen Ungleichungen, J. Reine Angew. Math. (1902), 127.
 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
(37 #5198)
 6.
J. H. Grace and A. Young, The algebra of invariants, University Press, Cambridge, 1903.
 7.
Philipp
Rothmaler, Some model theory of modules. II. On stability and
categoricity of flat modules, J. Symbolic Logic 48
(1983), no. 4, 970–985 (1984). MR 727787
(85e:03084), http://dx.doi.org/10.2307/2273662
 8.
Volker
Weispfenning, Elimination of quantifiers for certain ordered and
latticeordered abelian groups, Proceedings of the Model Theory
Meeting (Univ. Brussels, Brussels/Univ. Mons, Mons, 1980), 1981,
pp. 131–155. MR 620968
(82h:03022)
 9.
Volker
Weispfenning, Model theory of abelian 𝑙groups,
Latticeordered 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
(91h:03044), http://dx.doi.org/10.1007/BF01651327
 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
(12,352g)
 1.
 C. C. Chang and H. J. Keisler, Model Theory, NorthHolland Publishing Co., Amsterdam, 1973. MR 0409165 (53:12927)
 2.
 G. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, 1998. MR 1658673 (99g:90004)
 3.
 L. van den Dries, Some applications of a model theoretic fact to (semi)algebraic geometry, Nederl. Akad. Wetensch. Indag. Math. (1982), 397401. MR 0683527 (84m:14029)
 4.
 J. Farkas, Theorie der einfachen Ungleichungen, J. Reine Angew. Math. (1902), 127.
 5.
 R. W. Gilmer, Multiplicative ideal theory, Queen's Papers in Pure and Applied Mathematics, No. 12, Queen's University, Kingston, 1968. MR 0229624 (37:5198)
 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 (1983), 970985. MR 0727787 (85e:03084)
 8.
 V. Weispfenning, Elimination of quantifiers for certain ordered and latticeordered abelian groups, Bull. Soc. Math. Belg. Sér. B (1981), 131155. MR 0620968 (82h:03022)
 9.
 V. Weispfenning, Model theory of abelian groups. In A. M. W. Glass and W. C. Holland, eds., Latticeordered groups, Mathematics and its Applications, 48, Kluwer Academic Publishers, Dordrecht, 1989, pp. 4179. MR 1036073
 10.
 V. Weispfenning, Existential equivalence of ordered abelian groups with parameters, Arch. Math. Logic (1990), 237248. MR 1062728 (91h:03044)
 11.
 H. Weyl, The elementary theory of convex polyhedra. In H. W. Kuhn and A. W. Tucker, eds., Contributions to the Theory of Games, Volume I, Annals of Mathematics Studies, No. 24, Princeton University Press, Princeton, 1950, pp. 318. MR 0038088 (12:352g)
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Additional Information
Philip Scowcroft
Affiliation:
Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459
Email:
pscowcroft@wesleyan.edu
DOI:
http://dx.doi.org/10.1090/S000299470603978X
PII:
S 00029947(06)03978X
Received by editor(s):
July 19, 2004
Published electronically:
March 1, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
