Home Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
Article
Licensed
Unlicensed Requires Authentication

Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set

  • P. Douka EMAIL logo and V. Felouzis
Published/Copyright: August 17, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 3

Abstract

We give necessary and sufficient conditions for a family M of subsets of a set X which completely separates X to be a sublattice of the lattice of C(X,≤) of all convex subsets of X, with respect a suitable linear ordering ≤ of X. As an application we give a characterization of Hausdorff topological spaces which are orderable or suborderable.

MSC 2010: Primary 06B15; Secondary 06B30, 54F05
Keywords: lattices; linear ordered spaces; orderable topological spaces

(Communicated by Miroslav Ploščica)

Acknowledgement

The authors thank the anonymous referee whose valuable comments and suggestions led to substantial improvement of the paper.

The second author also gratefully acknowledges the hospitality and wishes to thank l’ Unité de Mathématiques Pures et Appliquées de l’ École Normale Supérieur de Lyon, where he was visiting during part of this work.

References

[1] Altwegg, M.: Zur Axiomatik der teilweise geordneten Mengen, Comment. Math. Helv. 24 (1950), 149–155.10.1007/BF02567030Search in Google Scholar

[2] Čech, E.: Topological Spaces, Wiley, New York, 1966.Search in Google Scholar

[3] Bludov, V. V.—Droste, M.—Glass, A. M. W.: Automorphism groups of totally ordered sets: a retrospective survey, Math. Slovaca 61 (2011), 373–388.10.2478/s12175-011-0018-1Search in Google Scholar

[4] Dalen, J. van—Wattel, E. A topological characterization of ordered spaces, Gen. Topology Appl. 3 (1973), 347–354.10.1016/0016-660X(73)90022-6Search in Google Scholar

[5] Deàk, E.: Theory and Applications of Directional Structures. Topics in Topology (Proc. Colloq., Keszthely, 1972). Colloq. Math. Soc. János Bolyai 8, North-Holland, Amsterdam, 1974.Search in Google Scholar

[6] Fishburn, P.: Betweenness, order and interval graphs, J. Pure Appl. Algebra 1 (1971), 159–178.10.1016/0022-4049(71)90016-8Search in Google Scholar

[7] Glivenko, V.: Geométrie des systèmes de choses normées, Amer. J. Math. 58 (1936), 799–828.10.2307/2371251Search in Google Scholar

[8] Glivenko, V.: Contributions a l’ étude des systèmes de choses normées, Amer. J. Math. 59 (1937), 941–956.10.2307/2371360Search in Google Scholar

[9] Hashimoto, J.: Betweenness geometry, Osaka Math. J. 10 (1958), 147–158.Search in Google Scholar

[10] Hedlíková, J.—Katriňák, T.: On a characterization of lattices by the betweenness relation on a problem of M. Kolibiar, Algebra Universalis 28 (1991), 389–400.10.1007/BF01191088Search in Google Scholar

[11] Hedlíková, J.—Katriňák, T.: Lattice betweenness relation and a generalization of Königs Lemma, Math. Slovaca 46 (1996), 343–354.Search in Google Scholar

[12] Huntington, E. V.—Kline, J. R.: Sets of independent postulates for betweenness with proof of complete independence, Trans. Amer. Math. Soc. 26 (1915), 6–24.Search in Google Scholar

[13] Kay, D. C.—Womble, E. W.: Axiomatic convexity theory and relationships between the Caratheodory, Helly, and Radon numbers, Pacific J. Math. 38 (1971), 471–485.10.2140/pjm.1971.38.471Search in Google Scholar

[14] Klaučová, O.: Characterization of distributive multilattices by a betweenness relation, Math. Slovaca 26 (1976), 119–129.Search in Google Scholar

[15] Kolibiar, M.: Charakterisierung der Verbände durch die Relation “zwischen”, Z. Math. Logik Glundlangen Math. 4 (1958), 89–100.10.1002/malq.19580040702Search in Google Scholar

[16] Lutzer, D. J.: On generalized ordered spaces, Dissertationes Math. (Rozprawy Mat.) 89 (1971).Search in Google Scholar

[17] Mendris, R.—Zlatoš, P.: Axiomatization and undecidability results for metrizable betweenness relations, Math. Slovaca 46 (1999), 305–313.10.1090/S0002-9939-1995-1219728-7Search in Google Scholar

[18] Mendris, R.—Zlatoš, P.: Axiomatization and undecidability results for linear betweenness relations, Proc. Amer. Math. Soc. 123 (1995), 873–882.10.2307/2160813Search in Google Scholar

[19] Menger, K.: Untersuschungen über die allgemeine Metrik, Math. Ann. 100 (1928), 75–163.10.1007/BF01448840Search in Google Scholar

[20] Peano, G.: I principii di geometria, Turin, 1889.Search in Google Scholar

[21] Peano, G.: Sui fondamenti dellia geometria, Riv. Math. 4 (1894), 51–90.Search in Google Scholar

[22] Pitcher, E.—Smiley, M. F.: Transititivities of Betweenness, Trans. Amer. Math. Soc. 52 (1942), 95–114.10.2307/1990155Search in Google Scholar

[23] Pasch, M.: Vorlesungen über neuere Geometrie, Teunberg, Leipzig, 1882.Search in Google Scholar

[24] Ploščica, M.: On a characterization of distributive lattices by the betweenness relation, Algebra Universalis 35 (1996), 249–255.10.1007/BF01195499Search in Google Scholar

[25] Renyi, A.: On random generating elements of a finite Boolean algebra, Acta Sci. Math. (Szeged) 22 (1961), 75–81.Search in Google Scholar

[26] Sholander, M.: Trees, lattices, order and betweenness, Proc. Amer. Math. Soc. 3 (1952), 369–381.10.1090/S0002-9939-1952-0048405-5Search in Google Scholar

[27] Šimko, J.: Linear and R-linear betweenness spaces, Math. Slovaca 51 (2001), 365–370.Search in Google Scholar

[28] Tarski, A.: What is an Elementary Geometry? In: The Axiomatic Method with Special Reference to Geometry and Physics. Proceedings of an International Symposium Held at the Univ. of Calif., Berkeley, Dec. 26, 1957, Jan. 4, 1958 (L. Henkin, P. Suppes, A. Tarski, eds.). Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1958, pp. 16–29.Search in Google Scholar

[29] Veblen, O.: A system of axioms for geometry, Trans. Amer. Math. Soc. 5 (1904), 343–384.10.1090/S0002-9947-1904-1500678-XSearch in Google Scholar

Received: 2013-4-24
Accepted: 2013-12-13
Published Online: 2016-8-17
Published in Print: 2016-6-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
https://doi.org/10.1515/ms-2015-0156
Keywords for this article
lattices; linear ordered spaces; orderable topological spaces

Articles in the same Issue

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 19.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0156/html?lang=en
Scroll to top button