Abstract
In 1983 A. Quilliot published his original work on graphs and ordered sets viewed as metric spaces. His approach was revolutionary. It was the first time that metric ideas and concepts could be defined in discrete sets. In particular one can show that graphs or order preserving maps are exactly the class of nonexpansive mappings defined on metric spaces. Pouzet and his students Jawhari and Misane were able to build on Quilliot’s ideas to establish some new insights into absolute retracts in ordered sets. For example it was amazing that the metric results discovered by Aronszajn and Panitchpakdi (Pac. J. Math. 6:405–439, 1956), the work of Isbell (Comment. Math. Helv. 39:439–447, 1964), and the fixed point theorems of Sine (Nonlinear Anal. 3:885–890, 1979) and Soardi (Proc. Am. Math. Soc. 73:25–29 1979) are exactly the Banaschewski–Bruns theorem (Archiv. Math. Basel 18:369–377, 1967), the MacNeille completion (Trans. Am. Soc. 42:416–460, 1937) and the famous Tarski fixed point theorem (Pac. J. Math. 5:285–309, 1955). Recently Abu-Sbeih and Khamsi used the same ideas to define a concept similar to externally hyperconvex metric sets introduced by Aronszajn and Panitchpakdi in their original work in ordered sets. They also proved a an intersection property similar to the one discovered by Baillon in metric spaces to show a common fixed point result. In conclusion this approach supports the idea that certain concepts of infinistic nature, like those which inspired metric spaces, can easily translate into discrete structures like ordered sets and graphs. In this chapter, we will only focus on ordered sets.
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References
Aronszajn N., Panitchpakdi, P.: Extensions of uniformly continuous transformations and hyperconvex metric spaces. Pac. J. Math. 6, 405–439 (1956)
Abu-Sbeih, M.Z., Khamsi, M.A.: On externally complete subsets and common fixed points in partially ordered sets. Fixed Point Theor. Appl. 2011, Article No. 97 (2011)
Aksoy, A.G., Khamsi, M.A.: A selection theorem in metric trees. Proc. Am. Math. Soc. 134, 2957–2966 (2006)
Baillon, J.B.: Nonexpansive mapping and hyperconvex spaces. In: Brown, R.F. (ed.) Fixed Point Theory and Its Applications, Contemporary Math., vol. 72, pp. 11–19. Am. Math. Soc., Providence (1988)
Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeille completion. Archiv. Math. Basel 18, 369–377 (1967)
Blumenthal, L.M.: Distance Geometries: A Study of the Development of Abstract Metrics. University of Missouri Studies, vol. 13, 1938
Blumenthal, L.M.: Theory and Applications of Distance Geometries. Claremont Press, Oxford (1953)
Blumenthal, L.M., Menger, K.: Studies in Geometry. W.H.Freeman and Co., San Francisco (1970)
Brodskii, M.S., Milman, D.P.: On the center of a convex set. Dokl. Akad. Nauk SSSR 59, 837–840 (1948) (Russian)
Caire, L., Cerruti, U.: Fuzzy relational spaces. Rendiconti Semin. della Facolta Sci. dell’Univ. Cagliari 47, 63–87 (1977)
Cerruti, U., Hohle, U.: Categorical foundations of probabilistic microgeometry. Séminaire de Mathématique Floue, Lyon, 189–246 (1983–1984)
Chen, J., Li, Z.: Common fixed points for Banach operator pairs in best approximation. J. Math. Anal. Appl. 336, 1466–1475 (2007)
De Marr, R.: Common fixed points for commuting contraction mappings. Pac. J. Math. 13, 1139–1141 (1963)
Dhage, B.C.: Generalized metric spaces and mappings with fixed point. Bull. Calcutta Math. Soc. 84, 329–336 (1992)
Duffus, D., Rival, I.: A structure theory for ordered sets. J. Discrete Math. 35, 53–118 (1981)
Espínola, R., Khamsi, M.A.: Introduction to hyperconvex spaces. In: Kirk, W.A., Sims, B. (eds.) Handbook of Metric Fixed Point Theory, pp. 391–435. Kluwer Academic, Dordrecht (2001)
Fréchet, M.: Les espaces Abstraits. Gauthier-Villars, Paris (1928)
Grätzer, G.A.: Universal Algebra. Springer, New York (1979)
Hell, P.: Absolute retracts of graphs. Lect. Notes math. 406, 291–301 (1974)
Higgs, D.: Injectivity in the topos of complete Heyting algebra valued sets. Canad. J. Math. 36, 550–568 (1984)
Higman, G.: Ordering by divisibilty in abstract algebra. Proc. Lond. Math. Soc. 3, 326–336 (1952)
Isbell, J.R.: Six theorems about injective metric spaces. Comment. Math. Helv. 39, 439–447 (1964)
Jawhari, E., Misane, D., Pouzet, M.: Retracts: graphs and ordered sets from the metric point of view. Contemp. Math. 57, 175–226 (1986)
Khamsi, M.A., Kreinovich, V., Misane, D.: A new method of proving the existence of Answer sets for disjunctive logic programs: A metric fixed point theorem for multivalued maps. Proc. of the Workshop on Logic Programming with Incomplete Information, Vancouver, British-Columbia, Canada, 1993
Khamsi, M.A., Misane, D.: Fixed point theorems in logic programming. Ann. Math. Artif. Intell. 21, 231–243 (1997)
Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004–1006 (1965)
Lawvere, F.W.: Metric spaces, generalized logic and closed categories. Rendiconti Semin. Math. Fisico Milano 43, 135–166 (1974)
MacNeille, H.: Partially ordered sets. Trans. Am. Soc. 42, 416–460 (1937)
Menger, K.: Untersuchungen über allgemeine Metrik. Math. Ann. 100, 75–163 (1928)
Menger, K.: Geométrie générale. Mémorial des Sciences mathématiques, No 124, Paris, Gauthiers-Villars, 1954
Mustafa, Z., Sims, B.: Some remarks concerning D-metric spaces. Proc. of International Conference on Fixed Point Theory and applications, pp. 189–198. Yokohama Publishers, Valencia Spain (2004)
Mustafa, Z., Sims, B.: A new approach to a generalized metric spaces. J. Nonlinear Convex Anal. 7, 289–297 (2006)
Penot, J.P.: Fixed point theorem without convexity. Bull. Soc. Math. France Mem. 60, 129–152 (1979)
Quilliot, A.: Homomorphismes, point fixes, rétractions et jeux de poursuite dans les graphes, les ensembles ordonnés et les espaces métriques. Thèse de Doctorat d’Etat, Univ. Paris VI, 1983
Quilliot, A.: An application of the Helly property to the partially ordered sets. J. Combin. Theor. A 35, 185–198 (1983)
Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc. 30, 264–286 (1930)
Sine, R.: On nonlinear contractions in sup-norm spaces. Nonlinear Anal. 3, 885–890 (1979)
Soardi, P.: Existence of fixed points of nonexpansive mappings in certain Banach lattices. Proc. Am. Math. Soc. 73, 25–29 (1979)
Takahashi, W.: A convexity in metric space and nonexpansive mappings. I. Kodai Math. Sem. Rep. 22, 142–149 (1970)
Tarski, A.: A lattice theoretical fixpoint theorem and its applications. Pac. J. Math. 5, 285–309 (1955)
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Abu-Sbeih, M.Z., Khamsi, M.A. (2014). Fixed Point Theory in Ordered Sets from the Metric Point of View. In: Almezel, S., Ansari, Q., Khamsi, M. (eds) Topics in Fixed Point Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-01586-6_6
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