Abstract

We show analogues of the classical Krein-Milman theorem for several ordered algebraic structures, especially in a semilattice (non-linear) framework. In that case, subsemilattices are seen as convex subsets, and for our proofs we use arguments from continuous lattice theory and abstract convexity theory.

Author information

Contact details are reproduced from the original publication and may be historical.

Paul Poncet

CMAP – École Polytechnique, Route de Saclay, 91128 Palaiseau, France and: INRIA, Saclay–Île-de-France

poncet@cmap.polytechnique.fr

Keywords

  • Abstract convexity
  • max-plus convexity
  • tropical convexity
  • Krein-Milman theorem
  • convex geometries
  • antimatroids
  • partially ordered sets
  • semilattices
  • Lawson semilattices
  • lattices

Mathematics Subject Classification

22A26; 52A01, 06A06, 06A12, 06B30, 14T05

References

  1. C. D. Aliprantis, K. C. Border: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd Ed., Springer, Berlin (2006).Google Scholar
  2. K. A. Baker: A Krein–Milman theorem for partially ordered sets, Amer. Math. Mon. 76 (1969) 282–283.https://doi.org/10.1080/00029890.1969.12000195Google Scholar
  3. H. Bauer: Minimalstellen von Funktionen und Extremalpunkte. II, Arch. Math. 11 (1960) 200–205.https://doi.org/10.1007/BF01236933Google Scholar
  4. J. Berman, G. Bordalo: Finite distributive lattices and doubly irreducible elements, Discrete Math. 178(1–3) (1998) 237–243.https://doi.org/10.1016/S0012-365X(97)81832-8Google Scholar
  5. G. Bińczak, A. B. Romanowska, J. D. H. Smith: Poset extensions, convex sets, and semilattice presentations, Discrete Math. 307(1) (2007) 1–11.https://doi.org/10.1016/j.disc.2006.09.021Google Scholar
  6. G. Birkhoff: Lattice Theory, American Mathematical Society Colloquium Publications 25, American Mathematical Society, New York (1948).Google Scholar
  7. G. Birkhoff, M. K. Bennett: The convexity lattice of a poset, Order 2(3) (1985) 223–242.https://doi.org/10.1007/BF00333128Google Scholar
  8. T. S. Blyth: Lattices and Ordered Algebraic Structures, Universitext, Springer, London (2005).Google Scholar
  9. N. Bourbaki: Éléments de Mathématique. Espaces Vectoriels Topologiques. Chapitres 1 à 5, Masson, Paris (1981).https://doi.org/10.1007/978-3-032-12156-1_1Google Scholar
  10. W. Briec, C. D. Horvath, A. M. Rubinov: Separation in B-convexity, Pac. J. Optim. 1(1) (2005) 13–30.Google Scholar
  11. P. Butkovič, H. Schneider, S. N. Sergeev: Generators, extremals and bases of max cones, Linear Algebra Appl. 421(2-3) (2007) 394–406.https://doi.org/10.1016/j.laa.2006.10.004Google Scholar
  12. T. H. Choe: Intrinsic topologies in a topological lattice, Pacific J. Math. 28 (1969) 49–52.https://doi.org/10.2140/pjm.1969.28.49Google Scholar
  13. T. H. Choe: On compact topological lattices of finite dimension, Trans. Amer. Math. Soc. 140 (1969) 223–237.https://doi.org/10.1090/S0002-9947-1969-0240012-1Google Scholar
  14. G. Cohen, S. Gaubert, J.-P. Quadrat: Duality and separation theorems in idempotent semimodules, Linear Algebra Appl. 379 (2004) 395–422.https://doi.org/10.1016/j.laa.2003.08.010Google Scholar
  15. M. Develin, B. Sturmfels: Tropical convexity, Doc. Math., J. DMV 9 (2004) 1–27 (electronic).https://doi.org/10.4171/dm/154Google Scholar
  16. A. Edalat: Dynamical systems, measures, and fractals via domain theory, Inf. Comput. 120(1) (1995) 32–48.https://doi.org/10.1006/inco.1995.1096Google Scholar
  17. P. H. Edelman: Meet-distributive lattices and the anti-exchange closure, Algebra Univers. 10(3) (1980) 290–299.https://doi.org/10.1007/BF02482912Google Scholar
  18. P. H. Edelman, R. E. Jamison: The theory of convex geometries, Geom. Dedicata 19(3) (1985) 247–270.https://doi.org/10.1007/BF00149365Google Scholar
  19. J. W. Ellis: A general set-separation theorem, Duke Math. J. 19 (1952) 417–421.https://doi.org/10.1215/S0012-7094-52-01941-8Google Scholar
  20. M. Erné: Bigeneration in complete lattices and principal separation in ordered sets, Order 8(2) (1991) 197–221.https://doi.org/10.1007/BF00383404Google Scholar
  21. M. Erné, M. Gehrke, A. Pultr: Complete congruences on topologies and down-set lattices, Appl. Categ. Struct. 15(1–2) (2007) 163–184.https://doi.org/10.1007/s10485-006-9054-3Google Scholar
  22. K. Fan: On the Krein-Milman theorem, in: Convexity (Washington, 1961), V. L. Klee (ed.), Proc. Sympos. Pure Math., 7, Amer. Math. Soc., Providence (1963) 211–219.https://doi.org/10.1090/pspum/007/0154097Google Scholar
  23. S. P. Franklin: Some results on order-convexity, Amer. Math. Mon. 69(5) (1962) 357–359.https://doi.org/10.1080/00029890.1962.11989897Google Scholar
  24. S. Gaubert, R. D. Katz: The Minkowski theorem for max-plus convex sets, Linear Algebra Appl. 421(2–3) (2007) 356–369.https://doi.org/10.1016/j.laa.2006.09.019Google Scholar
  25. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott: Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications 93, Cambridge University Press, Cambridge (2003).https://doi.org/10.1017/CBO9780511542725Google Scholar
  26. J. Goubault-Larrecq: A cone theoretic Krein–Milman theorem, Rapport de Recherche LSV-08-18, ENS Cachan, France (2008).Google Scholar
  27. S. Helbig: On Carathéodory’s and Kreĭn–Milman’s theorems in fully ordered groups, Commentat. Math. Univ. Carol. 29(1) (1988) 157–167.Google Scholar
  28. K. H. Hofmann, J. D. Lawson: Irreducibility and generation in continuous lattices, Semigroup Forum 13(4) (1976/77) 307–353.https://doi.org/10.1007/BF02194952Google Scholar
  29. K. H. Hofmann, A. Stralka: The algebraic theory of compact Lawson semilattices. Applications of Galois connections to compact semilattices, Diss. Math. (Rozprawy Mat.) 137 (1976) 54 p.Google Scholar
  30. C. D. Horvath, J. V. Llinares Ciscar: Maximal elements and fixed points for binary relations on topological ordered spaces, J. Math. Econ. 25(3) (1996) 291–306.https://doi.org/10.1016/0304-4068(95)00732-6Google Scholar
  31. R. E. Jamison: A General Theory of Convexity, PhD Thesis, University of Washington, Seattle (1974).Google Scholar
  32. R. E. Jamison: Some intersection and generation properties of convex sets, Compos. Math. 35(2) (1977) 147–161.Google Scholar
  33. R. E. Jamison-Waldner: Copoints in antimatroids, in: Proc. 11th Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1980), Vol. II, F. O. Hadlock et al. (ed.), Congress. Numer. 29, Utilitas Math., Winnipeg (1980) 535–544.Google Scholar
  34. R. E. Jamison-Waldner: A convexity characterization of ordered sets, in: Proc. 10th Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1979), F. Hoffman et al. (ed.), Congress. Numer. 24, Utilitas Math., Winnipeg (1979) 529–540.Google Scholar
  35. R. E. Jamison-Waldner: Partition numbers for trees and ordered sets, Pac. J. Math. 96(1) (1981) 115–140.https://doi.org/10.2140/pjm.1981.96.115Google Scholar
  36. R. E. Jamison-Waldner: A perspective on abstract convexity: classifying alignments by varieties, in: Convexity and Related Combinatorial Geometry (Norman, 1980), D. C. Kay et al. (ed.), Lecture Notes in Pure and Appl. Math. 76, Dekker, New York (1982) 113–150.Google Scholar
  37. S. Kakutani: Ein Beweis des Satzes von M. Eidelheit über konvexe Mengen, Proc. Imp. Acad. Jap. 13(4) (1937) 93–94.https://doi.org/10.3792/pia/1195579980Google Scholar
  38. K. Keimel: Topological cones: functional analysis in a T0 -setting, Semigroup Forum 77(1) (2008) 109–142.https://doi.org/10.1007/s00233-008-9078-0Google Scholar
  39. V. L. Klee, Jr.: Extremal structure of convex sets, Arch. Math. 8 (1957) 234–240.https://doi.org/10.1007/BF01899998Google Scholar
  40. M. Krein, D. Milman: On extreme points of regular convex sets, Stud. Math. 9 (1940) 133–138.https://doi.org/10.4064/sm-9-1-133-138Google Scholar
  41. M. Lassak: A general notion of extreme subset, Compos. Math. 57(1) (1986) 61–72.Google Scholar
  42. J. D. Lawson: Vietoris Mappings and Embeddings of Topological Semilattices, PhD Thesis, University of Tennessee (1967).Google Scholar
  43. J. D. Lawson: Topological semilattices with small semilattices, J. Lond. Math. Soc., II. Ser. 1 (1969) 719–724.https://doi.org/10.1112/jlms/s2-1.1.719Google Scholar
  44. J. D. Lawson: The relation of breadth and codimension in topological semilattices. II, Duke Math. J. 38 (1971) 555–559.https://doi.org/10.1215/S0012-7094-71-03868-3Google Scholar
  45. J. D. Lawson: Intrinsic topologies in topological lattices and semilattices, Pac. J. Math. 44 (1973) 593–602.https://doi.org/10.2140/pjm.1973.44.593Google Scholar
  46. J. D. Lawson: Joint continuity in semitopological semigroups, Ill. J. Math. 18 (1974) 275–285.https://doi.org/10.1215/ijm/1256051229Google Scholar
  47. J. D. Lawson, M. Mislove, H. A. Priestley: Infinite antichains in semilattices, Order 2(3) (1985) 275–290.https://doi.org/10.1007/BF00333134Google Scholar
  48. J. W. Lea, Jr.: Continuous lattices and compact Lawson semilattices, Semigroup Forum 13(4) (1976/77) 387–388.https://doi.org/10.1007/BF02194956Google Scholar
  49. J. R. Liukkonen, M. W. Mislove: Measure algebras of locally compact semilattices, in: Recent Developments in the Algebraic, Analytical, and Topological Theory of Semigroups (Oberwolfach, 1981), K. H. Hofmann et al. (ed.), Lecture Notes in Math. 998, Springer, Berlin (1983) 202–214.https://doi.org/10.1007/BFb0062030Google Scholar
  50. J. Martinez: Unique factorization in partially ordered sets, Proc. Amer. Math. Soc. 33 (1972) 213–220.https://doi.org/10.1090/S0002-9939-1972-0292723-5Google Scholar
  51. D. Milman: Characteristics of extremal points of regularly convex sets, Dokl. Akad. Nauk SSSR, II. Ser. 57 (1947) 119–122.Google Scholar
  52. B. Monjardet: A use for frequently rediscovering a concept, Order 1(4) (1985) 415–417.https://doi.org/10.1007/BF00582748Google Scholar
  53. B. Monjardet, R. Wille: On finite lattices generated by their doubly irreducible elements, Discrete Math. 73 (1989) 163–164.https://doi.org/10.1016/0012-365X(88)90144-6Google Scholar
  54. L. Nachbin: Topology and Order, Van Nostrand Mathematical Studies 4, D. Van Nostrand, Princeton (1965).Google Scholar
  55. S. Park: Comments on abstract convexity structures on topological spaces, Nonlinear Anal. 72(2) (2010) 549–554.https://doi.org/10.1016/j.na.2009.06.100Google Scholar
  56. P. Poncet: Infinite-Dimensional Idempotent Analysis: The Role of Continuous Posets, PhD Thesis, École Polytechnique, Palaiseau (2011).Google Scholar
  57. I. Singer: Abstract Convex Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York (1997).Google Scholar
  58. A. R. Stralka: Locally convex topological lattices, Trans. Amer. Math. Soc. 151 (1970) 629–640.https://doi.org/10.2307/1995517Google Scholar
  59. M. L. J. van de Vel: Lattices and semilattices: a convex point of view, in: Continuous Lattices and Their Applications (Bremen, 1982), R.-E. Hoffmann et al. (ed.), Lecture Notes in Pure and Appl. Math. 101, Dekker, New York (1985) 279–302.https://doi.org/10.1201/9781003072621-17Google Scholar
  60. M. L. J. van de Vel: A selection theorem for topological convex structures, Trans. Amer. Math. Soc. 336(2) (1993) 463–496.https://doi.org/10.1090/S0002-9947-1993-1169083-9Google Scholar
  61. M. L. J. van de Vel: Theory of Convex Structures, North-Holland Mathematical Library 50, North-Holland, Amsterdam (1993).Google Scholar
  62. N. The Vinh: Matching theorems, fixed point theorems and minimax inequalities in topological ordered spaces, Acta Math. Vietnam. 30(3) (2005) 211–224.Google Scholar
  63. A. D. Wallace: A fixed-point theorem, Bull. Amer. Math. Soc. 51 (1945) 413–416.https://doi.org/10.1090/S0002-9904-1945-08367-0Google Scholar
  64. A. Wieczorek: Spot functions and peripherals: Krein–Milman type theorems in an abstract setting, J. Math. Anal. Appl. 138(2) (1989) 293–310.https://doi.org/10.1016/0022-247X(89)90291-6Google Scholar
  65. A. Wirth: Some Krein–Milman theorems for order-convexity, J. Austral. Math. Soc. 18 (1974) 257–261.https://doi.org/10.1017/S1446788700022825Google Scholar

Suggested citation

P. Poncet. “Convexities on Ordered Structures Have Their Krein-Milman Theorem.” Journal of Convex Analysis 21 (2014), No. 1, 89–120. https://doi.org/10.68381/jca21005

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