Extending the idea of compressed algebra to arbitrary socle-vectors

Fix a codimension $r$ and a socle-vector $s$. Is there an (entry by entry) maximal $h$-vector $h$ among the $h$-vectors of all the (standard graded artinian) algebras having data $(r,s)$? Extending a definition of Iarrobino, if such an $h$ exists, we define as {\it generalized compressed} (GCA, in brief) any algebra having the data $(r,s,h)$. The two main results of this paper are: Theorem A, where we supply a very natural upper-bound $H$ for all the $h$-vectors possible for a given pair $(r,s)$; Theorem B, which asserts that, under certain conditions on the pair $(r,s)$, the upper-bound $H$ of Theorem A is actually achieved by a GCA. In particular, it follows that, when $r=2$, there always exists a GCA (having $h$-vector $H$). Moreover, we show that, in general, the hypotheses of Theorem B cannot be improved, i.e., under weaker conditions on the pair $(r,s)$, the upper-bound $H$ above is not always achieved.