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 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 .

**[1]**Garrett Birkhoff,*Lattice theory*, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053****[2]**Ivan Rival,*Maximal sublattices of finite distributive lattices*, Proc. Amer. Math. Soc.**37**(1973), 417–420. MR**0311527**, 10.1090/S0002-9939-1973-0311527-9**[3]**Ivan Rival,*Lattices with doubly irreducible elements*, Canad. Math. Bull.**17**(1974), 91–95. MR**0360387**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0360393-5

Keywords:
Finite distributive lattice,
sublattice,
maximal proper sublattice,
join-irreducible,
length

Article copyright:
© Copyright 1974
American Mathematical Society