We consider the set Γ(n) of all period sets of strings of length n over a finite alphabet. We show that there is redundancy in period sets and introduce the notion of an irreducible period set. We prove that Γ(n) is a lattice under set inclusion and does not satisfy the Jordan- Dedekind condition.We propose the first enumeration algorithm for Γ(n) and improve upon the previously known asymptotic lower bounds on the cardinality of Γ(n). Finally, we provide a new recurrence to compute the number of strings sharing a given period set.