Order

, Volume 17, Issue 2, pp 129–139 | Cite as

Collection of Finite Lattices Generated by a Poset

  • Branimir Šešelja
  • Andreja Tepavčević
Article

Abstract

It is proved that the collection of all finite lattices with the same partially ordered set of meet-irreducible elements can be ordered in a natural way so that the obtained poset is a lattice. Necessary and sufficient conditions under which this lattice is Boolean, distributive and modular are given.

finite distributive lattice finite lattice meet-irreducible representation of lattices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Behrendt, G. (1988) Maximal antichains in partially ordered sets, Ars. Combin. C 25, 149–157.Google Scholar
  2. 2.
    Bergman, G. M. and Zimmermann-Huisgen, B. (1990) Infinite joins are finitely joinirreducible, Order 7, 27–40.Google Scholar
  3. 3.
    Birkhoff, G. (1967) Lattice Theory, 3rd edn, Amer. Math. Soc., Providence, RI.Google Scholar
  4. 4.
    Bordalo, G. and Monjardet, B. (1996) Reducible classes of finite lattices, Order 13, 379–390.Google Scholar
  5. 5.
    Dilworth, R. P. (1960) Some combinatorial problems on partially ordered sets, in Combinatorial Analysis, Proc. Symp. Appl. Math. X, Amer. Math. Soc., pp. 85–90.Google Scholar
  6. 6.
    Erné, M. (1991) Bigeneration in complete lattices and principal separation in ordered sets, Order 8, 197–221.Google Scholar
  7. 7.
    Farley, J. D. (1996) Priestley duality for order-preserving maps into distributive lattices, Order 13, 65–98.Google Scholar
  8. 8.
    Gehrke, M. (1994) Uniquely representable posets, in Papers on General Topology and Applications (Flushing, NY, 1992), Ann. New York Acad. Sci. 728, New York Acad. Sci., New York, pp. 32–40.Google Scholar
  9. 9.
    Higgs, D. (1986) Lattice of crosscuts, Algebra Universalis 23, 10–18.Google Scholar
  10. 10.
    Kemp, P. A. (1997) On the representation of partially ordered sets, Rend. Circ. Mat. Palermo 2(1), 119–122.Google Scholar
  11. 11.
    Koh, K. M. (1983) On the lattice of maximum-sized antichains of a finite poset, Algebra Universalis 17, 73–86.Google Scholar
  12. 12.
    Ploščica, M. (1995) A natural representation of bounded lattices, Tatra Mountains Math. Publ. 5, 75–88.Google Scholar
  13. 13.
    Šešelja, B. and Tepavčević, A. (1994) Representation of lattices by fuzzy sets, Inform. Sci. 79, 171–180.Google Scholar
  14. 14.
    Šešelja, B. and Tepavčević, A. (1998) On generation of finite posets by meet-irreducibles, Discrete Math. 186, 269–275.Google Scholar
  15. 15.
    Šešelja, B. and Tepavčević, A. (1996) On the collection of lattices determined by the same poset of meet-irreducibles, Novi Sad J. Math. 26, 11–19.Google Scholar
  16. 16.
    Zádori, L. (1992) Order varieties generated by finite posets, Order 8, 341–348.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Branimir Šešelja
    • 1
  • Andreja Tepavčević
    • 1
  1. 1.Fac. of Sci., Institute of MathematicsUniversity of Novi SadNovi SadYugoslavia

Personalised recommendations