A commutative 1-monoid for classification with fuzzy attributes

In this paper the properties of an algebraic fuzzy structure are investigat ed in detail. The structure is suitable for modeling classifications throug h 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 classificat ion problems, e.g., fuzzy assessment of students, user modeling for fuzzy h ypermedia systems, spaces of the cognitive states of the user of a tutoring system, financial investments, medical diagnoses. The problem of getting t he unknown classification beginning from the final classification is deeply investigated and it is shown that the problem is strictly related to the s olution of an equation in the monoid. Thus it is possible to construct proc edures of the type 'what happens if which permit to attain significant resu lts both on the theoretical side and the applicative one. Finally, by means of this approach, both the absolute and the relative relevance of an attri bute are defined and evaluated, given a universe of discourse and a set of classifications. Moreover, this couple of features allow to develop a sophi sticated analysis of how a new attribute can be obtained beginning from a s et of attributes. (C) 2001 Elsevier Science Inc. All rights reserved.