Covering of minimal solutions to fuzzy relational equations

A minimal solution to a fuzzy relational equation is, loosely speaking, a solution such that the values of all its elements cannot be decreased. To find some minimal solution is simple – one can just identify the greatest solution to the equation and then lower the values of all its elements as much as possible. However, finding all minimal solutions is significantly more difficult since when decreasing the values one needs to go through all combinations of the elements. The paper shows that when considering an n-element scale of truth values equipped with Łukasiewicz or Goguen operations, the set of all minimal solutions can be represented by constrained solutions which allows to skip a considerable amount of these combinations.