Abstract.
A pair (x,y) of elements in a lattice is mismatching if x is join-irreducible, y is meet-irreducible, and xy. The excess of a lattice L is defined by ex(L):=|L|− min{|V x |+|I y |:(x,y) is mismatching}, where V x (I y ) is the principal filter (ideal) generated by x(y), respectively. In the first half of this paper, it is shown that a lattice has excess zero (one) if and only if it is isomorphic to a Boolean lattice (one of a chain of length two, a diamond, and a pentagon), respectively. In the second half, we show that for each relatively complemented lattice which is not Boolean, its size is at most five times its excess. Moreover we determine extremal configurations.
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Received: September 2, 1999 Final version received: July 25, 2000
Acknowledgments. The author thanks the referee for constructive comments.
Present address: Bureau of Human Resources, National Personnel Authority, 1-2-3 Kasumigaseki, Chiyoda-ku, Tokyo 100-8913, Japan. e-mail: abetet@jinji.go.jp
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Abe, T. Excess of a Lattice. Graphs Comb 18, 395–402 (2002). https://doi.org/10.1007/s003730200029
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DOI: https://doi.org/10.1007/s003730200029