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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

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