Publication:
Characterizing isometries on the order polytope with an application to the theory of fuzzy measures

Loading...
Thumbnail Image
Full text at PDC
Publication Date
2010
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
In this paper we study the group of isometries over the order polytope of a poset. We provide a result that characterizes any isometry based on the order structure in the original poset. From this result we provide upper bounds for the number of isometries over the order polytope in terms of its number of connected components. Finally, as an example of application, we recover the set of isometries for the polytope of fuzzy measures and the polytope of p-symmetric measures when the indifference partition is fixed.
Description
Keywords
Citation
G. Birkhoff, Lattice Theory, AMS Colloquium Publications, third ed., vol. 25, American Mathematical Society, 1967. T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005. B. Bollobás, G. Brightwell, A. Sidorenko, Geometrical techniques for estimating numbers of linear extensions, European Journal of Combinatorics 20 (5) (1999) 329–335. G. Brightwell, P. Winkler, Counting linear extensions is # P-complete. In: STOC ’91: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, New York, NY, USA, 1991, ACM, pp. 175–181. U.K. Chakraborty, Genetic and evolutionary computing, Information Sciences 178 (23) (2008) 4419–4420 (Including Special Section: Genetic and Evolutionary Computing). G. Choquet, Theory of capacities, Annales de l’Institut Fourier (5) (1953) 131–295. E.F. Combarro, P. Miranda, Identification of fuzzy measures from sample data with genetic algorithms, Computers and Operations Research 33 (10) (2006) 3046–3066. E.F. Combarro, P. Miranda, Adjacency on the order polytope with applications to the theory of fuzzy measures, Fuzzy Sets and Systems, in press. G. de Cooman, E. Miranda, Symmetry of models versus models of symmetry, in: W. Harper, G. Wheeler (Eds.), Probability and Inference: Essays in Honor of H.E. Kyburg, King’s College Publications, 2007, pp. 67–149. R. Dedekind, Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler, Festschrift Hoch Braunschweig Gesammelte Werke II (1897) 103–148 (in German). A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, The Annals of Mathematical Statistics (38) (1967) 325–339. D. Denneberg, Non-additive Measures and Integral, Kluwer Academic, 1994. D. Dubois, H. Prade, Possibility Theory, Plenum Press, 1985. F. Durante, R. Mesiar, S. Saminger-Platz, Editorial to the special issue devoted to copulas, measures and integrals, Information Sciences 179 (17) (2009) 2861–2862 (Copulas, Measures and Integrals). V. Garg, Algorithmic combinatorics based on slicing posets, Theoretical Computer Science 359 (1–3) (2006) 200–213. V. Garg, N. Mittal, A. Sen, Using order in distributed computing, in: Proceedings of American Mathematical Society (AMS) National Meeting, San Antonio, Texas, USA, January 2006. M.Grabisch, k-Order additive discrete fuzzy measures, in: Proceedings of sixth International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Granada, Spain, 1996, pp. 1345–1350. M. Grabisch, T. Murofushi, M. Sugeno (Eds.), Fuzzy Measures and Integrals-Theory and Applications, Number40 in Studies in Fuzziness and Soft Computing, Physica-Verlag, 2000. T. Hungerford, Algebra, Springer, 1974. R. Jegou, R. Medina, L. Nourine, Linear space algorithm for on-line detection of global predicates, in: J. Desel (Ed.), Proceedings of the International Workshop on Structures in Concurrency Theory (STRICT ’95), 1995. J. Kahn, J. Han Kim, Entropy and sorting, in: STOC ’92: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, New York, NY, USA, 1992, ACM, pp. 178–187. A.N. Karkishchenko, Invariant fuzzy measures on a finite algebra, in: Proceedings of the North American Fuzzy Information Processing (NAPIF’96), Berkeley, June 1996, pp. 588–592. G. Koshevoy, Distributive lattices and product of capacities, Journal of Mathematical Analysis and Applications 219 (1998) 427–441. Jiri Matousek, Lectures on Discrete Geometry, Springer-Verlag, New York, Inc., Secaucus, NJ, USA, 2002. F. Mattern, Virtual time and global states of distributed systems, in: Parallel and Distributed Algorithms: Proceedings of the InternationalWorkshop on Parallel and Distributed Algorithms, Elsevier Science Publishers B.V., North-Holland, 1989, pp. 215–226. R. Mesiar, E. Pap, Aggregation of infinite sequences, Information Sciences 178 (12) (2008) 3557–3564. P. Miranda, E.F. Combarro, On the structure of some families of fuzzy measures, IEEE Transactions on Fuzzy Systems 15 (6) (2007) 1068–1081. P. Miranda, E.F. Combarro, P. Gil, Extreme points of some families of non-additive measures, European Journal of Operational Research 33 (10) (2006) 3046–3066. P. Miranda, M. Grabisch, p-Symmetric bi-capacities, Kybernetica 40 (4) (2004) 421–440. P. Miranda, M. Grabisch, P. Gil, p-symmetric fuzzy measures, International Journal of Uncertainty Fuzziness, and Knowledge-Based Systems 10 (Suppl.)(2002) 105–123. Y. Narukawa, V. Torra, Fuzzy measures and integrals in evaluations of strategies, Information Sciences 177 (21) (2007) 4686–4695. E. Quaeghebeur, G. deCooman, Extreme lower probabilities, Fuzzy Sets and Systems 159 (16) (2008) 2163–2175. G.C. Rota, On the foundations of combinatorial theory I. Theory of Möbius functions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete (2) (1964) 340–368. G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, New Jersey, USA, 1976. G. Sirbiladze, T. Gachechiladze, Restored fuzzy measures in expert decision-making, Information Sciences 169 (1–2)(2005) 71–95. R. Stanley, Two poset polytopes, Discrete and Computational Geometry 1 (1) (1986) 9–23. G. Steiner, An algorithm for generating the ideals of a partial order, Operations Research Letters 5 (1986) 317–320. M. Sugeno, Theory of Fuzzy Integrals and its Applications, PhD Thesis, Tokyo Institute of Technology, 1974.
Collections