On fuzzy relational equations and the covering problem

Multi-adjoint algebras versus non-commutative residuated structures

“…for all x, y ∈ R. This operator is a u-norm if w 1 , w 2 > 0 and 0 < r < ∞ [32]. If we consider w 1 = w 2 = 0.5 and r = 1, then we have that M 0.5,0.5,1 (x, y) = 0.5x + 0.5 y, for all x, y ∈ R and this u-norm cannot have adjoint implications, since the conjunctor of an adjoint triple satisfies that ⊥ & y = ⊥ (Proposition 20 (2)) and, in this case, M 0.5,0.5,1 (0, y) = y 2 = 0, for all y ∈ R \ {0}.…”

“…Nonetheless, these operators were considered with the main purpose of solving the fuzzy relation equations with u-norms introduced in [32], the authors so consider continuous u-norms and so, these operators have adjoint implications [38]. Therefore, these u-norms are a particular case of a conjunctor in an adjoint triple.…”

“…Therefore, these u-norms are a particular case of a conjunctor in an adjoint triple. As a consequence, the setting given in [32] is a particular case of the multi-adjoint relation equations as well.…”

Fuzzy relational equations and the covering problem

“…Lin [17] extended Markovskii's work to fuzzy relational equations with max-Archimedean-t-norm composition. Further, Lin [18] investigated fuzzy relational equations with u-norm and transformed the problem of solving a system of fuzzy relational equations into covering problem. In 2012, Shieh [26] developed an efficient algorithm for finding minimal coverings.…”

“…In 2012, Shieh [26] developed an efficient algorithm for finding minimal coverings. In fact, Lin [17,18] and Shieh [26] discussed the relations between the minimal solutions of the equations and the irredundant coverings. However, fuzzy relational equations with inf-implication composition have no minimal solutions, there are maximal solutions of the equations, in this paper, we discuss fuzzy relational equations with inf-implication composition on [0,1].…”