System of fuzzy relation equations with sup-* composition in semi-linear spaces: minimal solutions

The problem of solvability of a system of fuzzy relation equations with sup-_* composition is considered in semilinear vector spaces. Based on the fact that a complete set of solutions is determined by minimal solutions, we focused on characterization of them. At first, sets of all minimal solutions of a single equation have been described under different assumptions on an underlying algebra. Dependently on the ordering of the support set, either necessary or sufficient conditions, or criteria of being a minimal solution have been obtained. Then minimal solutions of a system are build from minimal solutions of single equations.