4. Problems

Problem E-1. Which of the partially ordered sets listed as examples in the handout on that topic are lattices?

In cases where the lattice operations have more familiar names, give those names.

Problem E-2. A partially ordered set $S$ is a join-semilattice if every two elements have a least upper bound.

(a) Show that a finite join-semilattice with bottom element is a lattice.

(b) Give an example of an infinite join-semilattice with bottom element that is not a lattice.

(c) Invent and prove a theorem for join-semilattices similar to Theorem .

Problem E-3. In a vector space $V$, the set of subspaces, ordered by inclusion, is a lattice. What are the meet and join operations, in more familiar terms?

Problem E-4. In a group $G$, the set of subgroups, ordered by inclusion, is a lattice. What is the meet, in more familiar terms? How can the join of two subgroups be described, in terms of elements?