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

Pouzet, Maurice

Publications du Département de mathématiques (Lyon), 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