Lower Semimodular Types of Lattices: Frankl's Conjecture Holds for Lower Quasi-Semimodular Lattices |
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| Authors: | Tetsuya Abe Bumpei Nakano |
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| Institution: | (1) Department of Systems Science, Tokyo Institute of Technology 4259 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan. e-mail: abe@iaws−20.ia.noda.sut.ac.jp, JP;(2) Department of Artificial Intelligence and Systems Science, Tokyo Institute of Technology, 4259, Nagatsuta, Midori-ku, Yokohama 226-8502, Japan e-mail: bnakano@nkn.dis.titech.ac.jp, JP |
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| Abstract: | We introduce a measure of how far a lattice L is from being lower semimodular. We call it the lower semimodular type of L. A lattice has lower semimodular type zero if and only if it is lower semimodular. In this paper we discuss properties of
the measure and we show that Frankl's conjecture holds for lower quasi-semimodular lattices: if a lattice L is lower quasi-semimodular then there is a join-irreducible element x in L such that the size of the principal filter generated by x is at most (|L|− 1) /2.
Revised: July 2, 1997 |
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