  Let T be a complete local (Noetherian) ring and let A be a local subring of T
such that the completion of A with respect to its maximal ideal is T. We
investigate the possible structures of the partially ordered set Spec(A).
Specifically, we explore the minimal prime ideals of A and their formal fibers,
the maximal chains of prime ideals in A, and the number of prime ideals in A
containing combinations of minimal prime ideals of A.
