Maximal sublattices of finite distributive lattices. II
Proc. Amer. Math. Soc. 44 (1974), 263-268
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Abstract: Let be a lattice, join-irreducible in and meet-irreducible in . As is well known the sets and play a central role in the arithmetic of a lattice of finite length and particularly, in the case that is distributive. It is shown that the ``quotient set'' plays a somewhat analogous role in the study of the sublattices of a lattice of finite length. If is a finite distributive lattice, its quotient set ) in a natural way determines the lattice of all sublattices of . By examining the connection between and , where is a maximal proper sublattice of a finite distributive lattice , the following is proven: every finite distributive lattice of order which contains a maximal proper sublattice of order also contains sublattices of orders , and ; and, every finite distributive lattice contains a maximal proper sublattice such that either or , where denotes the length of .
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