Which of the following lattices form the Boolean algebra?
Which of the following lattices form the Boolean algebra?
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Answer: The cube-shaped (8-element) Hasse diagram forms the Boolean algebra.
First diagram (six-vertex cycle): Not Boolean. An element in this cycle has two different complements, so complements are not unique. In a Boolean algebra complements must exist for every element and be unique; the presence of multiple complements and the resulting failure of distributivity rule this out.
Second diagram (diamond with an extra top making a pentagon-like shape): Not Boolean. This lattice contains a sublattice isomorphic to the N5/pentagon configuration, which violates distributivity. Even if some complements exist, the lack of distributivity prevents the lattice from being Boolean.
Third diagram (a chain of four elements): Not Boolean. Internal elements of this chain do not have complements, so not every element has a complement. A Boolean algebra requires every element to have a complement, so this chain cannot be Boolean.
Fourth diagram (cube-shaped Hasse diagram): Boolean. This 8-element lattice is isomorphic to the powerset lattice of a 3-element set ordered by inclusion. It is distributive and every element has a unique complement (the set complement), so it satisfies all requirements of a Boolean algebra.
Quick checklist to verify a Boolean algebra:
1) The structure is a lattice (every pair has join and meet).
2) The lattice is distributive (it contains no M3 or N5 sublattice).
3) Every element has a unique complement.