I am offering a reading course on posets. We will be following chapter 3 of Enumerative Combinatorics by Richard Stanley.

Homework 1 due Wednesday 4/9.

• Prove Proposition 3.3.3.
• Are the following statements right? If so, prove it. Otherwise, give a counter example.
  • For any infinite poset, there exists an infinite chain.
  • For any infinite poset, there is either an infinite chain or an infinite antichain.

• Second try to prove Proposition 3.3.3: (ii) implies (i). We need to show that for any interval [x,y], for any u in [x,y], we can find a w in [x,y] such that u meet w = x and u join w = y. It is clear when u = x or y. Assume u != x or y. Prove the following statements.
  • There exists w in [x,y] such that w != x and u meet w = x.
  • Let w be a maximal one satisfying the above condition, then u join w = y.
• Do a small example to verify Theorem 3.4.1.
• Problem 1 on page 312.

Homework 3 due Wednesday 4/23.

• Problem 7 on page 314.
• Problem 13 on page 314.

Homework 4 due Wednesday 4/30.

• Problem 32 on page 160. (Try not to look at the solution first.)

Homework 5 due Wednesday 5/7.

• Show that an interval [x, y] of a geometric lattice L is also a geometric lattice.
• Let x <= y in a geometric lattice L. Show that \mu(x,y) = \pm 1 if and only if the interval [x, y] is isomorphic to a boolean algebra.

Homework 6 due Wednesday 5/14.

• Problem 4 on page 313.
• Problem 9 on page 314.

Homework 7 due Wednesday 5/21.

• Read sections 3.12, 3.13 and Theorem 4.1 in the paper Supersolvable lattices by Stanley, which is the solution to problem 49/a on page 163.