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Math Help - Which posets are lattices

  1. #1
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    Which posets are lattices

    Could someone help with this problem?

    Determine which of the following posets (S, <=) are lattices.

    1. A = {1, 3, 6, 9, 12} and <= is the divisibility relation.
    2. B = {1, 2, 3, 4, 5} and <= is the divisibility relation.
    3. C = {1, 5, 25, 100} and <= is the divisibility relation.
    4. Both 1 and 3

    My answer: 3

    Is this right? I think it is but I'm not 100% sure.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Which posets are lattices

    Yes you are right
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  3. #3
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    Re: Which posets are lattices

    for divisibility posets, meet is a gcd, and join is a lcm.

    #1) 6 and 9 have no join (6 divides 12 but 9 does not)
    #2) 2 and 3 have no join, neither do 2 and 5, 3 and 4, or 4 and 5.

    #3) is a linear order, that is for all a,b in P: {a∧b,a∨b} = {a,b}. this is not only a lattice, but a complete, distributive one.
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