Algebraic analysis of fuzzy systems

In this paper the authors have developed an algebraic theory, suitable for the analysis of fuzzy systems. They have used the notions of semiring and semimodule, introduced the notion of semilinear space and given numerous examples of them and defined also the notions of linear dependence and independence. Then, they 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 the applications.