In this paper the properties of an algebraic fuzzy structure are investigated in detail. The structure is suitable for modeling classi®cations through clusters composed of conventional sets and fuzzy attributes. We show that the structure is an integral commutative l-monoid. The expressive power of the structure is such that several situations can be viewed as classi®cation problems, e.g., fuzzy assessment of students, user modeling for fuzzy hypermedia systems, spaces of the cognitive states of the user of a tutoring system, ®nancial investments, medical diagnoses. The problem of getting the unknown classi®cation beginning from the ®nal classi®cation is deeply investigated and it is shown that the problem is strictly related to the solution of an equation in the monoid. Thus it is possible to construct procedures of the type `what happens if' which permit to attain signi®cant results both on the theoretical side and the applicative one. Finally, by means of this approach, both the absolute and the relative relevance of an attribute are de®ned and evaluated, given a universe of discourse and a set of classi®cations. Moreover, this couple of features allow to develop a sophisticated analysis of how a new attribute can be obtained beginning from a set of attributes.