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Semimodular lattices with isomorphic graphs

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Abstract

Two discrete modular lattice ℒ and ℳ have isomorphic graphs if and only if ℒ is of the form A × ℬ and ℳ is of the form A × ℬ for some lattices A and ℬ and . We prove that for discrete semimodular lattices ℒ and ℳ this latter condition holds if and only if ℒ and ℳ have isomorphic graphs and the isomorphism preserves the order on all cover-preserving sublattices of ℒ which are isomorphic to the seven-element, semimodular, nonmodular lattice (see Figure 1). This answers in the affirmative a question posed by J. Jakubik.

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Authors and Affiliations

  1. Department of Mathematics, La Trobe University, 3083, Bundoora, Australia

    C. Ratanaprasert & B. A. Davey

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  1. C. Ratanaprasert
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  2. B. A. Davey
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Communicated by I. Rival

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Ratanaprasert, C., Davey, B.A. Semimodular lattices with isomorphic graphs. Order 4, 1–13 (1987). https://doi.org/10.1007/BF00346648

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  • DOI: https://doi.org/10.1007/BF00346648

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