Let Am = {1, 2, 3, . . . , m}. We naturally order the numbers in Am with , which is a partial ordering. We may order elements in Am An by (a, b) (c, d) if and only if a c and b d. For example in A4, A3, (1, 2) (2, 3), but (4, 3) (3, 2).

a) Prove that defines a partial ordering on Am An.

b) Draw the Hasse diagram for (A3 A3, ).

c) Is (A3 A3, ) a complemented lattice? Explain.

d) Is (A3 A3, ) a distributed lattice? Explain.

e) Is (A3 A3, ) a Boolean lattice? Explain.