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Sublattices of Modular Lattices of Finite Length

Published online by Cambridge University Press:  20 November 2018

Ivan Rival
Ivan Rival*
Affiliation:
Department of Math.The University of Manitoba Winnipeg, Canada R3T 2N2
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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].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 1 , April 1975 , pp. 95 - 98
Copyright
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