Fuzzy equations max − * with conditionally cancellative operations

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.