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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Maximal sublattices of finite distributive lattices. II


Author: Ivan Rival
Journal: Proc. Amer. Math. Soc. 44 (1974), 263-268
MSC: Primary 06A35
MathSciNet review: 0360393
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Abstract: Let $ L$ be a lattice, $ J(L) = \{ x \in L\vert x$ join-irreducible in $ L\} $ and $ M(L) = \{ x \in L\vert x$ meet-irreducible in $ L\} $. As is well known the sets $ J(L)$ and $ M(L)$ play a central role in the arithmetic of a lattice $ L$ of finite length and particularly, in the case that $ L$ is distributive. It is shown that the ``quotient set'' $ Q(L) = \{ b/a\vert a \in J(L),b \in M(L),a \leqq b\} $ plays a somewhat analogous role in the study of the sublattices of a lattice $ L$ of finite length. If $ L$ is a finite distributive lattice, its quotient set $ Q(L)$) in a natural way determines the lattice of all sublattices of $ L$. By examining the connection between $ J(K)$ and $ J(L)$, where $ K$ is a maximal proper sublattice of a finite distributive lattice $ L$, the following is proven: every finite distributive lattice of order $ n \geqq 3$ which contains a maximal proper sublattice of order $ m$ also contains sublattices of orders $ n - m,2(n - m)$, and $ 3(n - m)$; and, every finite distributive lattice $ L$ contains a maximal proper sublattice $ K$ such that either $ \vert K\vert = \vert L\vert - 1$ or $ \vert K\vert \geqq 2l(L)$, where $ l(L)$ denotes the length of $ L$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0360393-5
PII: S 0002-9939(1974)0360393-5
Keywords: Finite distributive lattice, sublattice, maximal proper sublattice, join-irreducible, length
Article copyright: © Copyright 1974 American Mathematical Society