Algebraic analysis of fuzzy systems

Authors:
Antonio Di Nola;Ada Lettieri;Irina Perfilieva;Vilém Novák
Affiliations:
Universitá di Salerno, Facoltá di Scienze, Dipt. di Matematica e Informatica, Via S. Allende, 84081 Baronissi, Salerno, Italy;Universitá di Napoli, Dipt. di Costruzioni e Metodi Matematici in Architettura, Via Monteoliveto 3, 80134 Napoli, Italy;University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic;University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic
Venue:
Fuzzy Sets and Systems
Year:
2007

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Abstract

In this paper, we have developed an algebraic theory, suitable for the analysis of fuzzy systems. We have used the notions of semiring and semimodule, introduced the notion of semilinear space, have given numerous examples of them and defined also the notions of linear dependence and independence. We have shown that the composition operation, which plays an essential role in the analysis of fuzzy systems because of its role in the compositional rule of inference, can be interpreted as a homomorphism between special semimodules. Consequently, this operation is, in a certain sense, a linear operation. This property formally explains why fuzzy systems are attractive for applications.