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maximal chains in finite distributive lattices

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  • karincvetkovah
    Hi! Consider a finite distributive lattice L. One may assume that L is a sublattice of a power set of some finite set. Denote by j1,..., j_r all with join
    Message 1 of 1 , Jul 1, 2005
      Hi!

      Consider a finite distributive lattice L. One may assume that L is a
      sublattice of a power set of some finite set. Denote by j1,..., j_r
      all with join irreducible elements of L; assume that they are
      ordered:

      1. singletons come first, then two element sets,...

      2. singletons are ordered (for example as natural numbers); m-element
      subsets are ordered lexicographically.

      From all sequances of the form

      j1, j1 V j2, ..., j1 V j2 V ...j_{r-1} V j_r
      j1, j1 V j2, ..., j1 V j2 V ...j_r V j_{r-1}
      ....
      j_r, j_r V j_{r-1}, ..., j_r V j_{r-1} V ... V j1

      exclude those that are not chains and obtain all maximal chains in L,
      say C1,...,C_k.

      Is it true that L can be obtained as follows.

      1. Start with C_1

      2. L'=C_1 U... U C_{l_1}

      When adding the chain C_l, there is exactly one vertex on C_l that is
      not in L', and two edges (looking lattices as graphs.)

      Say, v in C_l, but not in L', and u,w in L' s.t. u<v<w in L.

      Since L' is connected, there exists a path from u to w in L. This
      path has length 2 because of the maximality of chains (is this always
      true?)

      Hence there exists v' in L' s.t. u<v'<w.

      Thank you, Karim
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