The Conditions That Fuzzy Relational Equations Have A Unique Minimal Solution

This paper investigates the conditions that fuzzy relational equations have minimal solutions. It first discusses how to determine whether a fuzzy relational equation with sup-product composition has a unique minimal solution (resp. a unique solution) in [0,1], it mainly uses the characteristic matrix to give the determinant method that the fuzzy relational equation has a unique solution (resp. a unique solution). It then verifies the conditions that a fuzzy relational equation with sup-conjunctor composition has a unique minimal solution (resp. a unique solution) in [0,1]. It primarily shows two operators Lτ and Lτ and presents some necessary and sufficient conditions that the equation has a unique minimal solution (resp. a unique solution). Finally, this paper investigates minimal solutions of fuzzy relational equations with sup-U composition on complete lattices. When the domain is a finite set and the right hand side is a join-irreducible element, it mainly gives a condition that the fuzzy relational equation has minimal solutions.