Discussion
Consider the following statements :
(I) Every distributive lattice is complemented and complementation is unique.
(II) Every modular lattice is complemented and complementation is unique
Then which of the following is true ?
Answer: C
Neither (I) nor (II) is true.
Not every distributive lattice is complemented: the three-element chain \(0 \lt a \lt 1\) is distributive, but \(a\) has no complement (no \(b\) with \(a \wedge b = 0\) and \(a \vee b = 1\)). Distributivity only guarantees that complements are unique when they exist.
The same chain is modular, so (II) fails for the same reason. Modularity does not even give uniqueness: in the diamond \(M_{3}\) each atom has two complements.
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