Fuzzy equations max − * with conditionally cancellative operations

Abstract

Fuzzy systems of equations A∘x=b are studied, where A∈[0, 1]m×n, b∈[0, 1]m and ∘ stands for max−* product. In the case of continuous, strictly increasing triangular norm*, B.S. Shieh (Information Sciences 177(19) (2007) 4208–4215) showed that non-zero coordinates of minimal solutions are coordinates of the greatest solution. The goal of this paper is a presentation of the weakest assumptions about the operation* which guarantee such result.Thus, the fuzzy system of equations is considered for a left-continuous, increasing, conditionally cancellative operation* such that 1*0=0. Minimal solutions of the system are described as orthogonal projections of the greatest solution and the number of minimal solutions of A∘x=b is estimated. Moreover, an efficient method is presented for the determination of the family of all minimal solutions.