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Sublattices of Modular Lattices of Finite Length
Published online by Cambridge University Press: 20 November 2018
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It is well-known that the join-irreducible elements J(L) and the meet-irreducible elements M(L) of a lattice L of finite length play a central role in its arithmetic and, especially, in the case that L is distributive. In [3] it was shown that the quotient set Q(L) = {b/a | a ∊ J(L), b ∊ M(L), a ≤ b} plays a somewhat analogous role in the study of the sublattices of L. Indeed, in a lattice L of finite length, if S is a sublattice of L then S = L — ∪b/a∊A [a, b] for some A ⊆ Q(L). Furthermore, the converse actually characterizes finite distributive lattices [3].
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- Copyright © Canadian Mathematical Society 1975
References
1.
Birkhoff, G., Lattice Theory, 3rd ed., American Mathematical Society, Providence, R.I., 1967.Google Scholar
2.
Rival, I., Maximal Sublattices of Finite Distributive Lattices, Proc. Amer. Math. Soc.
37 (1973), 417-420.Google Scholar
3.
Rival, I., Maximal Sublattices of Finite Distributive Lattices (II), Proc. Amer. Math. Soc.
44 (1974), 263-268.Google Scholar
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