The following is the Hasse diagram of the poset [{a, b, c, d, e}, <=]: the top element a covers the three incomparable elements b, c and d, and each of b, c, d covers the bottom element e (a 'diamond' with three middle atoms). The poset is: A. not a lattice B. a lattice but not a distributive lattice C. a distributive lattice but not a Boolean algebra D. a Boolean algebra
GATE 2005 · Discrete Mathematics · Lattice · medium
Answer: The poset is a lattice (M3) but fails the distributive law, hence is not distributive and not Boolean: the answer is B. a lattice but not a distributive lattice.
Show it is a lattice: any two distinct atoms (e.g. b, c) have join a and meet e; an atom with a gives join a and meet that atom; an atom with e gives join that atom and meet e. Every pair has a unique lub and glb, and a top a and bottom e exist, so the poset is a lattice. Option A is wrong.
Test the distributive law: compute the left side: c join d = a, so b meet (c join d) = b meet a = b. Compute the right side: b meet c = e and b meet d = e, so (b meet c) join (b meet d) = e join e = e. Since b != e, the distributive law fails, so the lattice is not distributive. Option C is wrong.
Rule out Boolean and conclude: a Boolean algebra must be distributive, but we just showed this lattice is not distributive, so it cannot be Boolean. Option D is wrong. The only consistent description is 'a lattice but not a distributive lattice'.