Question

v The partition lattice Πn is not distributive for n ≥ 3. Explain why.

Answer 1

Why partition lattice is not distributive for n≥3

alternate method : why partition lattice is distributive for n≥3.

We know that By the Fundamental Theorem on Distributive Lattices, every distributive lattice is isomorphic to the lattice of lower order ideals of some poset P ordered by inclusion.

The Boolean lattice Bn is the distributive lattice corresponding to the poset P with n elements and no relations . P has an automorphism with a cycle of length 2.

For n ≥ 3, the set partition lattice Πn has an automorphism with a cycle of length 2. So the partional lattice has property of Boolean lattice for n≥3. So partition lattice is distributive for n≥3 .