Are there "NP-Intermediate-Complete" problems? Assume P $\ne$ NP.

Ladner's Theorem says that there are NP Intermediate problems (problems in NP that are neither in P nor NP-Complete).  I have found some veiled references online that suggest (I think) that there are many "levels" of mutually reducible languages within NPI that definitely do not all collapse into one.

I have some questions about the structure of these levels.


Are there "NP-Intermediate-Complete" problems - that is, NP-Intermediate problems to which every other NP-Intermediate problem is polytime reducible?
Sort NP - P into equivalence classes, where mutual reducibility is the equivalence relation.  Now impose an ordering on these equivalence classes: $A &gt; B$ if the problems in $B$ reduce to problems in $A$ (so clearly the NP-Complete equivalence class is the maximum element).  Is this a total ordering (i.e. the problems are arranged in an infinite descending chain)?  If not, does the "tree structure" of the partial ordering have a finite branching factor?
Are there any other interesting known structural components of NP - P?  Are there any interesting open questions about the underlying structure?


If any of these are currently unknown, I would be interested to hear that as well.

Thanks!
I don't really have references for these results--they aren't hard to prove once you understand Ladner's theorem.


No, for any NP-incomplete set A there is another set B strictly between A and SAT.
These equivalence classes are known as polynomial-many-one degrees. You can embed any finite poset into the degrees below NP. In particular the degrees are not totally ordered or finitely branching. 
This all depends on what you mean by "interesting". There is a huge theory of the degree structure of the computable sets (see Soare's book for instance) and many of those questions have not been ported down to polynomial-time sets. For instance, can you have NP sets A and B whose join is equivalent to SAT and whose meet is equivalent to the empty set?