Maximal sublattices of finite distributive lattices. II
Author:
Ivan Rival
Journal:
Proc. Amer. Math. Soc. 44 (1974), 263268
MSC:
Primary 06A35
MathSciNet review:
0360393
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Abstract: Let be a lattice, joinirreducible in and meetirreducible 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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Garrett
Birkhoff, Lattice theory, Third edition. American Mathematical
Society Colloquium Publications, Vol. XXV, American Mathematical Society,
Providence, R.I., 1967. MR 0227053
(37 #2638)
 [2]
Ivan
Rival, Maximal sublattices of finite
distributive lattices, Proc. Amer. Math.
Soc. 37 (1973),
417–420. MR 0311527
(47 #89), http://dx.doi.org/10.1090/S00029939197303115279
 [3]
Ivan
Rival, Lattices with doubly irreducible elements, Canad. Math.
Bull. 17 (1974), 91–95. MR 0360387
(50 #12837)
 [1]
 G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R.I., 1967. MR 37 #2638. MR 0227053 (37:2638)
 [2]
 I. Rival, Maximal sublattices of finite distributive lattices, Proc. Amer. Math. Soc. 37 (1973), 417420. MR 0311527 (47:89)
 [3]
 , Lattices with doubly irreducible elements, Canad. Math. Bull. (to appear). MR 0360387 (50:12837)
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DOI:
http://dx.doi.org/10.1090/S00029939197403603935
PII:
S 00029939(1974)03603935
Keywords:
Finite distributive lattice,
sublattice,
maximal proper sublattice,
joinirreducible,
length
Article copyright:
© Copyright 1974
American Mathematical Society
