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Math Help - Pairs of lattices

  1. #1
    Member Goku's Avatar
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    Pairs of lattices

    Find all pairs of lattices (L,K)(up to isomorphism) such that

    a) L x K contains 20 elements
    b) L is non-distributive
    c) K has at least 3 elements
    d) the greatest element of L x K covers precisely 4 elements.

    From what I understand from this question K can have 4, 5, 10, 20
    elements, but since L is non-distributive it has to be at least 5 elements since it has to be isomorphic to M3 and N5.
    So K has to be 4 elements.

    How do you find these lattices???
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  2. #2
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    Re: Pairs of lattices

    By some element x covering four other elements I am assuming the question means that there are four elements immediately under x, with no intermediate elements. Is it true that if greatest element of one lattice covers m elements and the greatest element of another covers n elements, then the greatest element of the product covers mn elements? If so, then L cannot be M3...
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  3. #3
    Member Goku's Avatar
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    Re: Pairs of lattices

    I do not think that it will cover mn elements (this is just a guess)...

    Take L= M3 and M to be the chain 4 this L x K covers 4 elements... maybe it covers m+n elements...

    Any ideas...
    Last edited by Goku; September 29th 2011 at 07:54 AM.
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  4. #4
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    Re: Pairs of lattices

    You are right. If the maximum element a of M covers a_1,\dots,a_m and the maximum element b of N covers b_1,\dots,b_n, then (a,b) covers (a_1,b),\dots,(a_m,b), (a,b_1),\dots,(a,b_n). So, either L is M3 and the top element of K has 1 descendant, ot L is N5 and the maximum of K has 2 descendants.
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