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Maximal sublattices and Frattini sublattices of bounded lattices

Part of: Lattices

Published online by Cambridge University Press:  09 April 2009

M. E. Adams ,
Ralph Freese ,
J. B. Nation  and
Jürg Schmid
M. E. Adams
Affiliation:
Department of Mathematics and Computer Science Suny New Paltz NY 12561USA e-mail: adamsm@matrix.newpaltz.edu
Ralph Freese
Affiliation:
Department of Mathematics University of HawaiiHonolulu HI 96822USA e-mail: ralph@math.hawaii.edu jb@math.hawaii.edu
J. B. Nation
Affiliation:
University of BernSidlerstrasse 5 CH-3012 BernSwitzerland e-mail: schmid@math-stat.unibe.ch
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Abstract

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We investigate the number and size of the maximal sublattices of a finite lattice. For any positive integer k, there is a finite lattice L with more that ]L]k sublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

MSC classification

Secondary: 06B05: Structure theory 06B20: Varieties of lattices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 1 , August 1997 , pp. 110 - 127
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Abad, Manuel 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. and Sichler, J., ‘Frattini sublattices in varieties of lattices’, Colloq. Math. 44 (1982), 181–184.CrossRefGoogle Scholar
[4]Chen, C. C., Koh, K. M. and Tan, S. K., ‘On the Frattini sublattice of a finite distributive lattice’, Algebra Universalis 5 (1975), 88–97.CrossRefGoogle Scholar
[5]Day, A., ‘Splitting lattices generate all lattices’, Algebra Universalis 7 (1977), 163–170.CrossRefGoogle Scholar
[6]Day, A., ‘Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices’, Canad. J. Math. 31 (1979), 69–78.CrossRefGoogle Scholar
[7]Ježek, Freese J. and Nation, J. B., Free Lattices, volume 42 of Mathematical Surveys and Monographs (Amer. Math. Soc., Providence, 1995).Google Scholar
[8]Freese, R., Ježek, J. and Nation, J. B., ‘Lattices with large minimal extensions’, preprint, 1997.Google Scholar
[9]Ježek, J. and Slavík, V., ‘Primitive lattices’, Czechoslovak Math. J. 29 (1979), 595–634 Russian summary.CrossRefGoogle Scholar
[10]Koh, K. M., ‘On the Frattini sublattice of a lattice’, Algebra Universalis 1 (1971), 104–116.CrossRefGoogle Scholar
[11]McKenzie, R., ‘Equational bases and non-modular lattice varieties’, Trans. Amer. Math. Soc. 174 (1972), 1–43.CrossRefGoogle Scholar
[12]Nation, J. B., ‘An approach to lattice varieties of finite height’, Algebra Universalis 27 (1990), 521–543.CrossRefGoogle Scholar
[13]Rival, I., ‘Maximal sublattices of finite distributive lattices’, Proc. Amer Math. Soc. 37 (1973), 417–420.CrossRefGoogle Scholar