Increasing continuous operations in fuzzy max-⁎ equations and inequalities

In this paper, solutions of fuzzy systems of equations A ○ x = b and systems of inequalities A ○ x ≥ b , A ○ x ≤ b are considered, where A ∈ [ 0 , 1 ] m × n , b ∈ [ 0 , 1 ] m , ○ stands for max-⁎ product and x ∈ [ 0 , 1 ] n is an unknown vector. In this paper the families of all solutions are described. During the last thirty years this problem was considered by many authors. Our goal is to achieve the most general assumptions for which sets of solutionsʹ systems of equations and inequalities are the union of lattice intervals. Especially, a method of determination of all minimal solutions in such systems is indicated. As a result, the families of all solutions of fuzzy systems of inequalities A ○ x ≤ b and A ○ x ≥ b are described. As the main part of these results, the method of determination of the family of all minimal solutions in max-⁎ system of equations with an increasing operation ⁎, continuous on the second argument is presented.