Duality and separation theorems in idempotent semimodules

Linear Algebra and Its Applications

Volumn 379, Issue 1-3 SPEC. ISS, 2004, Pages 395-422

  Cohen, Guy ^a   Gaubert, Stéphane ^b   Quadrat, Jean Pierre ^b  

^a UNIVERSITÉ PARIS EST   (France)

^b INRIA ROCQUENCOURT   (France)

Author keywords

Column space; Dual pairs; Duality; Galois connection; Generalized conjugacies; Hahn Banach theorem; Linear extension; Max plus semiring; Projection; Residuation; Row space; Semimodules

Indexed keywords

APPROXIMATION THEORY; BOOLEAN FUNCTIONS; COMPUTATIONAL METHODS; CONFORMAL MAPPING; MATRIX ALGEBRA; SET THEORY; TOPOLOGY;

DUAL PAIRS; DUALITY; GALOIS CONNECTION; GENERALISED CONJUGACIES; HAHN-BANACH THEOREM; LINEAR EXTENSION; MAX-PLUS SEMIRING; RESIDUATION; ROW SPACES; SEMIMODULES;

THEOREM PROVING;

EID: 0742285656     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2003.08.010     Document Type: Conference Paper

References (51)

1. Akian M., Gaubert S., Kolokoltsov V. Invertibility of functional Galois connections. C.R. Acad. Sci. Paris Ser. I. 335:2002;1-6.
2. Aliprantis C.D., Border K.C. Infinite Dimensional Analysis. A Hitchiker's Guide. 1999;Springer.
3. Baccelli F., Cohen G., Olsder G.J., Quadrat J.P. Synchronization and Linearity - an Algebra for Discrete Event Systems. 1992;Wiley.
4. Birkhoff G. Lattice Theory. American Mathematical Society Colloquium Publications. vol. XXV:1940;AMS, Providence, RI.
5. Blyth T.S., Janowitz M.F. Residuation Theory. 1972;Pergamon Press.
6. Bourbaki N. Espaces Vectoriels Topologiques. Éléments de Mathématique. 1964;Livre V. Hermann.
7. Cao Z.Q., Kim K.H., Roush F.W. Incline Algebra and Applications. 1984;Ellis Horwood.
8. Carré B.A. An algebra for network routing problems. J. Inst. Math. Appl. 7:1971;273-294.
9. Carré B.A. Graphs and Networks. 1979;The Clarendon Press, Oxford University Press, New York.
10. Cohen G., Dubois D., Quadrat J.P., Viot M. A linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automat. Control. 30:1985;210-220.
11. Cohen G., Gaubert S., Quadrat J.P. Kernels, images and projections in dioids. Proceedings of WODES'96. 1996;IEE, Edinburgh.
12. Cohen G., Gaubert S., Quadrat J.P. Linear projectors in the max-plus algebra. 5th IEEE Mediterranean Conference on Control and Systems. 1997;Paphos, Cyprus.
13. Cohen G., Gaubert S., Quadrat J.P. Max-plus algebra and system theory: where we are and where to go now. Annu. Rev. Control. 23:1999;207-219.
14. Cohen G., Gaubert S., Quadrat J.P. Separation theorem for max-plus semimodules. Menaldi J.L., Rofman E., Sulem A. Proceedings of "Optimal Control and Partial Differential Equations" 2001;IOS Press.
15. G. Cohen, S. Gaubert, J.-P. Quadrat, I. Singer, Max-plus convex sets and functions, ESI, Vienna, 2003, Preprint 1341. Also arXiv:math.FA/0308166.
16. R.A. Cuninghame-Green, Process synchronization in a steelworks - a problem of feasibility, in: J. Banbury, J. Maitland (Eds.), Proceedings of the 2nd International Conference on Operations research, Aix-en-Provence, France, 1961.
17. Cuninghame-Green R.A. Describing industrial processes with interference and approximating their steady state behavior. Oper. Res. Quat. 13(1):1962;95-100.
18. Cuninghame-Green R.A. Minimax Algebra. Lecture Notes in Economics and Mathematical Systems. vol. 166:1979;Springer.
19. Cuninghame-Green R.A. Minimax algebra and applications. Adv. Imaging Electron Phys. 1995.
20. M.L. Dubreil-Jacotin, L. Lesieur, R. Croisot, Leçons sur la Théorie des Treillis, des Structures Algébriques Ordonnées, et des Treillis géométriques, vol. XXI of Cahiers Scientifiques, Gauthier Villars, Paris, 1953.
21. S. Gaubert, Théorie des systèmes linéaires dans les diödes, Thèse, École des Mines de Paris, 1992.
22. Gaubert S., Plus M. Methods and applications of (. max,+ ) linear algebra Reischuk R., Morvan M. STACS'97, LNCS. vol. 1200:1997;Springer, Lübeck.
23. Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S. A Compendium of Continuous Lattices. 1980;Springer.
24. Golan J.S. The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. vol. 54:1992;Longman Sci & Tech.
25. Gondran M., Minoux M. Valeurs propres et vecteurs propres dans les diödes et leur interprétation en théorie des graphes. EDF, Bulletin de la Direction des Etudes et Recherches, Serie C, Mathématiques Informatique. 2:1977;25-41.
26. Gondran M., Minoux M. Linear algebra in dioids: a survey of recent results. Ann. Discrete Math. 19:1984;147-164.
27. Gondran M., Minoux M. Graphes, Diödes et semi-anneaux. 2002;TEC & DOC, Paris.
28. Gunawardena J. Idempotency. 1998;Publications of the Newton Institute, Cambridge University Press.
29. Hasse M. Über die Behandlung graphentheoretischer Probleme unter Verwendung der Matrizenrechnung. Wiss. Z. Techn. Univ. Dresden. 10:1961;1313-1316.
30. Kim K.H. Boolean Matrix Theory and Applications. 1982;Marcel Dekker, New York.
31. V. Kolokoltsov, personal communication, 1999.
32. Kolokoltsov V., Maslov V. Idempotent Analysis and Applications. 1997;Kluwer Academic Publishers.
33. Korbut A.A. Extremal spaces. Dokl. Akad. Nauk SSSR. 164:1965;1229-1231.
34. Litvinov G.L., Shpiz G.B. Nuclear semimodules and kernel theorems in idempotent analysis: an algebraic approach. Dokl. Math. Sci. 2002;. Also math.FA/0206026.
35. Litvinov G.L., Maslov V.P., Shpiz G.B. Linear functionals on idempotent spaces: an algebraic approach. Dokl. Math. 58(3):1998;389-391.
36. Litvinov G.L., Maslov V.P., Shpiz G.B. Idempotent functional analysis: an algebraic approach. Math. Notes. 69(5):2001;696-729. Also reprint arXiv:math.FA/0009128.
37. V. Maslov, S. Samborskiǐ; (Eds.), Idempotent Analysis, Adv. in Sov. Math., vol. 13, AMS, RI, 1992.
38. V.P. Maslov, Méthodes Opératorielles, Mir. Moscou. French Transl. 1987, 1973.
39. Max Plus, Linear systems in ( max,+ )-algebra, in: Proceedings of the 29th Conference on Decision and Control, Honolulu, 1990.
40. P. Moller, Théorie algébrique des Systèmes à Événements Discrets, Thèse, École des Mines de Paris, 1988.
41. Romanovskiǐ I.V. Optimization of stationary control of discrete deterministic process in dynamic programming. Kibernetika. 3(2):1967;66-78.
42. Samborskiǐ S.N., Shpiz G.B. Convex sets in the semimodule of bounded functions. Idempotent Analysis. 1992;135-137 Amer. Math. Soc. Providence, RI.
43. Vorob'ev N.N. An extremal matrix algebra. Dokl. Akad. Nauk SSSR. 152:1963;24-27.
44. Vorob'ev N.N. Extremal algebra of positive matrices. Elektron. Informationsverarbeit. Kybernetik. 3:1967;39-71.
45. Vorob'ev N.N. Extremal algebra of non-negative matrices. Elektron. Informationsverarbeit. Kybernetik. 6:1970;303-311.
46. Wagneur E. Moduloids and pseudomodules. 1. Dimension theory. Discrete Math. 98:1991;57-73.
47. E. Wagneur, personal communication, 1991.
48. Yoeli M. A note on a generalization of Boolean matrix theory. Amer. Math. Monthly. 68:1961;552-557.
49. K. Zimmermann, Extremální Algebra, Ekonomický ùstav CSAV, Praha (in Czech), 1976.
50. Zimmermann K. A general separation theorem in extremal algebras. Ekonom.-Mat. Obzor. 13(2):1977;179-201.
51. Zimmermann U. Linear and Combinatorial Optimization in Ordered Algebraic Structures. 1981;North Hollan.