A commutative 1-monoid for classification with fuzzy attributes

In this paper the properties of an algebraic fuzzy structure are investigated in detail. The structure is suitable for modeling classifications through clusters composed of conventional sets and fuzzy attributes. We show that the structure is an integral commutative I-monoid. The expressive power of the structure is such that several situations can be viewed as classification 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, financial investments, medical diagnoses. The problem of getting the unknown classification beginning from the final classification 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 significant 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 defined and evaluated, given a universe of discourse and a set of classifications. 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. (C) 2001 Elsevier Science Inc. All rights reserved.