When Is There a Unique Socle-Vector Associated to a Given h -Vector?

First, we construct a bijection between the set of h -vectors and the set of socle-vectors of Artinian algebras. As a corollary, we find the minimum codimension that an Artinian algebra with a given socle-vector can have. Then, we study the main problem in the article: Determining when there is a unique socle-vector for a given h -vector. We solve the problem completely if the codimension is at most 3. [ABSTRACT FROM AUTHOR]