Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-06-08T15:39:14.437Z Has data issue: false hasContentIssue false
  • English
  • Français

Covering in the lattice of subuniverses of a finite distributive lattice

Part of: Lattices

Published online by Cambridge University Press:  09 April 2009

Zsolt Lengvárszky  and
George F. McNulty
Zsolt Lengvárszky
Affiliation:
Department of Computer Science, University of South Carolina, Columbia, SC 29208., USA e-mail: zlengvar@cs.sc.edu
George F. McNulty
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208.USA e-mail: mcnulty@math.sc.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The covering relation in the lattice of subuniverses of a finite distributive lattices is characterized in terms of how new elements in a covering sublattice fit with the sublattice covered. In general, although the lattice of subuniverses of a finite distributive lattice will not be modular, nevertheless we are able to show that certain instances of Dedekind's Transposition Principle still hold. Weakly independent maps play a key role in our arguments.

Keywords

distributive latticesublatticecoveringmaximal sublattice

MSC classification

Secondary: 06B05: Structure theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 3 , December 1998 , pp. 333 - 353
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Abad, M. and Adams, M. E., ‘The Frattini sublattice of a finite distributive lattice’, Algebra Universalis 32 (1994), 314–329.CrossRefGoogle Scholar
[2]Adams, M. E., ‘The Frattini sublattice of a distributive lattice’, Algebra Universalis 3 (1973), 216–228.CrossRefGoogle Scholar
[3]Adams, M. E., Dwinger, P. and Schmid, J., ‘Maximal sublattices of finite distributive lattices’, Algebra Universalis 36 (1996), 488–504.CrossRefGoogle Scholar
[4]Adams, M. E., Freese, R., Nation, J. B. and Schmid, J., ‘Maximal sublattices and Frattini sublattices of bounded lattices’, J. Austral. Math. Soc. (Series A) 63 (1997), 110–127.CrossRefGoogle Scholar
[5]Balbes, R. and Dwinger, P., Distributive Lattices (University of Missouri Press, Missouri, 1974).Google Scholar
[6]Birkhoff, G., ‘On the combination of subalgebras’, Proc. Cambridge Philos. Soc. 29 (1933). 441–464.CrossRefGoogle Scholar
[7]Birkhoff, G., Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25, 3rd edition (Amer. Math. Soc., Providence, RI, 1967).Google Scholar
[8]Chen, C. C., Koh, K. M. and Tan, S. K., ‘Frattini sublattices of distributive lattices’, Algebra Universalis 3 (1973), 294–303.CrossRefGoogle Scholar
[9]Czédli, G., Huhn, A. P. and Schmidt, E. T., ‘Weakly independent subsets in lattices’, Algebra Universalis 20 (1985), 194–196.CrossRefGoogle Scholar
[10]Davey, B. A. and Priestley, H. A., Introduction to Lattices and Order (Cambridge University Press, Cambridge, 1990).Google Scholar
[11]Hashimoto, J., ‘Ideal theory for lattices’, Math. Japon. 2 (1952), 149–186.Google Scholar
[12]Kuratowski, K., Topologie, I (Monogr. Mat., T. lII, Warsaw, 1933).Google Scholar
[13]Lengvárszky, Z., Independent Subsets in Lattices (Ph.D. Thesis, University of South Carolina, 1996).Google Scholar
[14]McKenzie, R., ‘Equational bases and non-modular lattice varieties’, Trans. Amer. Math. Soc. 174 (1972), 1–43.CrossRefGoogle Scholar
[15]Moore, E. H., Introduction to a form of general analysis, Amer. Math. Soc. Colloq. Publ. 2 (Amer. Math. Soc., Providence, RI, 1910).CrossRefGoogle Scholar
[16]Ore, O., ‘Combinations of closure operators’, Ann. Math. 44 (1943), 514–533.CrossRefGoogle Scholar
[17]Ore, O., ‘Some studies on closure relations’, Duke Math. J. 10 (1943), 761–785.CrossRefGoogle Scholar
[18]Rival, I., ‘Maximal sublattices of finite distributive lattices’, Proc. Amer. Math. Soc. 37 (1973), 417–420.CrossRefGoogle Scholar
[19]Rival, I., ‘Maximal sublattices of finite distributive lattices, II’, Proc. Amer. Math. Soc. 44 (1974), 263–268.CrossRefGoogle Scholar
[20]Ryter, C. and Schmid, J., ‘Deciding Frattini is NP-complete’, Order 11 (1994), 257–279.CrossRefGoogle Scholar
[21]Tůma, J., ‘On simultaneous representations of distributive lattices’, Acta Sci. Math. (Szeged) 58 (1993), 67–74.Google Scholar
[22]Vogt, F., ‘Bialgebraic contexts for finite distributive lattices’, Algebra Universalis 35 (1996), 151–165.CrossRefGoogle Scholar
[23]Ward, M., ‘The closure operators of a lattice’, Ann. Math. 43 (1942), 191–196.CrossRefGoogle Scholar