Faces of quasidensity

In this paper, we discuss the various structures associated with maximally monotone and quasidense subsets of the product of a real Banach space and its dual. In the first half of the paper, we give four statements equivalent to the assertion that a maximally monotone subset be quasidense, including the statement that the set be of ``type (NI)''. Then we consider the Fitzpatrick and Gossez extensions of a maximally monotone set to the dual. We give eight equivalent formulations for the Fitzpatrick extension of a quasidense maximally monotone set and prove that in, this case, the Fitzpatrick and Gossez extensions coincide. This leads to the result that a maximally monotone set is quasidense exactly when it is of ``type (D)''. We also give an example of a quasidense maximally monotone set whose Fitzpatrick extension, though maximally monotone, is not quasidense. In the final part of this paper, we prove that a maximally monotone set is quasidense if, and only if, it is of ``type (FP)''. The proof of ``only if'' is much simpler than the two which have been given up to now.