충청수학회지 (Journal of the Chungcheong Mathematical Society)

  • 제25권1호
  • /
  • Pages.105-113
  • /
  • 2012
  • /
  • 1226-3524(pISSN)
  • /
  • 2383-6245(eISSN)
충청수학회 (The Chungcheong Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

NOTES ON MODULAR ORDERED SETS

  • 발행 : 2012.02.15
⟨ 이전 논문 다음 논문 ⟩

초록

Generalizing modular lattices, a concept of modular ordered sets was introduced by Chajda and Rachunek. In this paper, we characterize modular ordered sets as those partially ordered set P satisfying that for $a,\;b,\;c\;{\in}\;P\;with\;b\;{\leq}\;c,\;l(a,\;b)\;=\;l(a,\;c)\;and\;u(a,\;b)\;=\;u(a,\;c)$ imply $b\;=\;c$. Using this, we obtain a sufficient condition for them. We also discuss the modularity of the Dedekind-MacNeille completions of ordered sets.

키워드

참고문헌

  1. G. Birkhoff, Lattice Theory, Ame. Math. Soc. Collq. Publ. Vol. 25, providence, R. L, 1967.
  2. I. Chajda and J. Rachunek, Forbidden Configurations for Distributive and Modular Ordered Sets, Order 5 (1989), 407-423. https://doi.org/10.1007/BF00353659
  3. B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1990.
  4. Y. H. Hong, S. J. Lim and S. H. Shin, Modularity and Distributivity of Ordered Sets, J. Nat. Sci. Sookmyung Women's Univ. 13 (2002), 83-87.
  5. M. Kolibiar, Modular Ordered sets, Math. Slovaca, 44 (1994), no. 2, 139-141.