Comparability graphs of lattices |
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Authors: | Jonathan David Farley Stefan E. Schmidt |
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Affiliation: | a Department of Applied Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States b Physical Science Laboratory, P.O. Box 30002, Las Cruces, NM 88003, United States |
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Abstract: | A theorem of N. Terai and T. Hibi for finite distributive lattices and a theorem of Hibi for finite modular lattices (suggested by R.P. Stanley) are equivalent to the following: if a finite distributive or modular lattice of rank d contains a complemented rank 3 interval, then the lattice is (d+1)-connected.In this paper, the following generalization is proved: Let L be a (finite or infinite) semimodular lattice of rank d that is not a chain (d∈N0). Then the comparability graph of L is (d+1)-connected if and only if L has no simplicial elements, where z∈L is simplicial if the elements comparable to z form a chain. |
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Keywords: | 06C10 06C05 05C40 06D99 05B25 05B35 06C20 06E99 |
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