Une approche métrique de la rétraction dans les ensembles ordonnés et les graphes

Pouzet, Maurice

Publications du Département de mathématiques (Lyon), Compte rendu des journées infinitistes, no. 2B (1985), pp. 59-89.

[1] N. Aronszajn, P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces. Pacific J. Math. 6 (1956), p. 405-439. | MR | Zbl

[2] K. Baclawski and A. Bjorner, Fixed points in partially ordered sets, Advances in Mathematics, 31 (1979), p. 263-287. | MR | Zbl

[3] B. Banaschewski and G. Bruns, Categorical characterization of the Mac Neille completion, Archiv. der Math. Basel 18 (1967), p. 369-377. | MR | Zbl

[4] L.M. Blumenthal, K. Menger, Studies in geometry 1970, W.H. Freeman and Co. San Francisco. | MR | Zbl

[5] U. Cerruti and U. Hohle, Categorical fundations of probabilistic microgeometry, Séminaire de "Mathématique floue" LYON (1983-1984) p. 189-246.

[6] D. Duffus and I. Rival, A structure theory for ordered sets, J. of Discrete Math. 35 (1981), P. 53-118. | MR | Zbl

[7] D. Duffus and M. Pouzet, Representing ordered sets by chains, in : Orders : Descriptions and Role (M. Pouzet and D. Richard, eds). Annals of Discrete Math. 23 (1984) p. 81-98. | MR | Zbl

[8] J. Elton, Pei-Kee-Lin, E. Odell and S. Szarek, Remarks on the fixed point problem for non expansive maps in Fixed Points and non expansive mappings (Sine Ed.) Contemporary Math. Vol. 18, (1983), p. 87-120. | MR | Zbl

[9] M. Frechet, Rend. Circ. Math. Palermo, Vol. 22 (1906), p. 6.

[10] M. Frechet, Les espaces abstraits, Paris 1928. | JFM

[11] F. Hausdorff, Grundzüge der Mengenlehre, 1914, Leipzig. | JFM

[12] P. Hell, Absolute retracts of graphs, Lecture notes 406 (1974) p. 291-301. | MR | Zbl

[13] P. Hell, Graph retractions, Atti dei conveigni lincei 17, teorie combinatorie (1976) p. 263-268. | MR | Zbl

[14] P. Hell, Rétractions de graphes. PhD. Université de Montréal, Juin 1972, 148 pages.

[15] P. Hell and I. Rival, Absolute retracts and varieties of reflexive graphs, preprint, 1983. | MR | Zbl

[16] D. Higgs, Injectivity in the topos of complete Heyting algebra valued sets, Canadian J. Of Math. 36 (1984) p. 550-568. | MR | Zbl

[17] G. Higman, Ordering by divisibility in abstract algebra, Proc. London Math. Soc. (3) 2 (1952) p. 326-336. | MR | Zbl

[18] R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964) p. 65-76. | MR | Zbl

[19] V.I. Istratescu, Fixed point theory, an introduction. Math. and its applications, Vol. 7 (1981) D. Reidel. | MR | Zbl

[20] E. Jawhari, M. Pouzet, I. Rival, A classification of reflexive graphs : The use of "holes". Rapport de recherche du Laboratoire d'Algèbre ordinale et algorithmique, Lyon (1983). To appear in Canadian J. of Math. | MR | Zbl

[21] E. Jawhari, Les rétractions dans les graphes. Applications et généralisations, Thèse de 3ème cucle, n° 1318 (Juillet 1983). Lyon.

[22] J.L. Kelley, General Topology, (1955), Van Nostrand. | MR | Zbl

[23] W. A. Kirk, Fixed point theory for non expansive mapping, Lecture notes in math., 886 (1981) p. 484-505. | MR | Zbl

[24] E.W. Kiss, L. Marki, P. Prohle and W. Tholen, Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity, Studia Scientiarum Mathematicarum Hungarica 18 (1983), p. 79-141. | MR | Zbl

[25] J.B. Kruskal, The theory of well quasi ordering : a frequently discovered concept, J. Comb. Th. (A) 13, p. 197-305/ | MR | Zbl

[26] H. Macneille, Partially ordered sets. Trans. Amer. Math. Soc 42 (1937) p. 416-460. | JFM | MR

[27] D. Misane, Retracts absolus d'ensembles ordonnés et de graphes. Propriété du point fixe. Thèse de doctorat de 3ème cycle, n° 1571 (Septembre 1984), Lyon.

[28] P. Nevermann and R. Wille, The strong selection property and ordered sets of finite length, Alg. Univ. 18 (1984) p. 18-28. | MR | Zbl

[29] R. Nowakowski and I. Rival, A fixed edge theorem for graphs with loops. J. Graph theory 3 (1979) p. 339-350. | MR | Zbl

[30] R. Nowakowski and I. Rival, The smallest graph variety containing all paths, J. of Discrete Math. 43 (1983) p. 223-234. | MR | Zbl

[31] E. Pesh, Minimal extension of graphs to absolute retracts, preprint n° 839, July 1984. Technische Hoschule Darmstadt. | Zbl

[32] M. Pons Valles, Contribucio a l'estudi d'estructures uniformes sobre conjunts ordenats, Thesis (1984), Barcelona.

[33] M. Pouzet, I. Rival, Every countable lattice is a retract of a direct product of chains, Alg. Univ. 18 (1984) p. 295-307. | MR | Zbl

[34] M. Pouzet, Retracts, recent and old results on graphs, ordered sets and metric spaces. Circulating manuscript, 29 pages, Nov. 1983. | Zbl

[35] A. Quilliot, Homomorphismes, points 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).

[36] A. Quilliot, An application of the Helly property to the partially ordered sets, J. Comb. Theory, série A, 35 (1983) p. 185-198. | MR | Zbl

[37] F.P. Ramsey, On a problem of formal logic. Proc London Math. Soc. 30, p. 264-286. | JFM | MR

[38] I. Rival, A fixed point theorem for finite partially ordered sets, J. of Comb. theory (1976) p. 309-318. | MR | Zbl

[39] I. Rival, R. Wille, The smallest order variety containing all chains, Discrete Math., 35, p. 203-212. | MR | Zbl

[40] Z. Semadeni, Banach spaces of continuous functions, Vol. I. Monografie Matematyczne, Warsawa (1971). | MR | Zbl

[41] R. Sine, On non linear contractions in Sup. norm. spaces. Non linear analysis, TMA, 3 (1979) p. 885-890. | MR | Zbl

[42] R. Sine, Fixed points and non expansive mappings (R. Sine ed.) Contemporary Math. Vol. 18. AMS. | Zbl

[43] D.R. Smart, Fixed point theorems, Cambridge tracts in Math. 66 (1974) Cambrdige University Press. | MR | Zbl

[44] P.M. Soardi, Existence of fixed points of non expansive mappings in certain banach lattices, Proc. A.M.S. (1979) p. 25-29. | MR | Zbl

[45] A. Tarski, A lattice theoretical fixed point theorem and its applications. Pacific J. Math. 5 (1955), p. 285-309. | MR | Zbl