## maximal chains in finite distributive lattices

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• 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.

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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