Residuation in fuzzy algebra and some applications
Applications of systems of linear equations over structures different from the classical field were studied in the 1960s and in recent years have again received increasing attention as a tool for modelling discrete event systems or fuzzy relations. Several authors have observed that a system of the form \(R \otimes x = b\) always has a `principal' solution \(x^* \), i.e. such that the system is solvable if and only if \(x^*\) is a solution; and in this case \(x^*\) is the maximum solution. The connection with residuation theory was pointed out by the first author [Minimax algebra (Lect. Notes Econ. Math. Syst. 166 (1979; Zbl 0399.90052)] and \textit{A. Di Nola} [Fuzzy Sets Syst. 34, No. 3, 365-376 (1990; Zbl 0701.04003)]. The aim of this paper is to show further implications of residuation theory for fuzzy algebra, including a method for testing linear dependence of sets of vectors. Next we address `the inverse problem' studied by \textit{W. Pedrycz} [Fuzzy Sets Syst. 49, No. 3, 339-355 (1992; Zbl 0805.04005)] and propose an efficient algorithm for its solution. Analogies of this approach with methods used in a similar algebra, where multiplication is a group operation, are stressed.
scientific article; zbMATH DE number 891004