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Math Help - length of Lattices

  1. #1
    Member Goku's Avatar
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    length of Lattices

    Prove that

    length(P x Q) = length(P) + length(Q)

    where P, Q are ordered sets of finite length.

    Any ideas of where to start...
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  2. #2
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    Re: length of Lattices

    What is the definition of the length of an ordered set? Are we talking about partial or total orders?
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  3. #3
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    Re: length of Lattices

    partially ordered sets, where the length is the the size of the longest chain in the poset.
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  4. #4
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    Re: length of Lattices

    If a_1,\dots,a_n and b_1,\dots,b_m are chains in P and Q, respectively, then (a_1,b_1),(a_2,b_1),\dots,(a_n,b_1), (a_n,b_2),\dots,(a_n,b_m) is a chain in P x Q, so length(P x Q) >= length(P) + length(Q). Conversely, if you have a chain (a_1,b_1)>\dots>(a_n,b_n) in P x Q, then for each i you have a_{i+1}>a_i or b_{i+1}>b_i (or both). So, you can construct chains in P and Q whose total length is >= n. This means that length(P) + length(Q) >= length(P x Q).
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