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A structure theorem for dismantlable lattices and enumeration

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Abstract

The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely, `A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for any adjunct representation of a dismantlable lattice the number of chains as well as the number of times a pair of elements occurs remains the same. If a dismantlable lattice L has n elements and n+k edges then it is proved that the number of irreducible elements of L lies between n-2k-2 and n-2. These results are used to enumerate the class of lattices with exactly two reducible elements, the class of lattices with n elements and upto n+1 edges, and their subclasses of distributive lattices and modular lattices.

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Authors and Affiliations

  1. Department of Mathematics, University of Pune, Pune, 411 007, India

    N. K. Thakare & B. N. Waphare

  2. Department of Mathematics, S.S.V.P.S's Late Karmveer Dr. P. R. Ghogrey Science College, Dhule, 424 005, India

    M. M. Pawar

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  1. N. K. Thakare
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  2. M. M. Pawar
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  3. B. N. Waphare
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Thakare, N.K., Pawar, M.M. & Waphare, B.N. A structure theorem for dismantlable lattices and enumeration. Periodica Mathematica Hungarica 45, 147–160 (2002). https://doi.org/10.1023/A:1022314517291

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