Fuzzy Relation Equations with Lower and Upper Semicontinuous Triangular Norms
In this Ch. we present another generalization of the fuzzy relation equations which have been considered in the previous chapters. Here we focus our attention on a broad class of logical connectives applied in fuzzy set theory, i.e. triangular norms (for short, t-norms) and conorms (for short, s-norms) which in turn enable us to consider max-t and min-s compositions. In Sec.8.1 we present essentially the concept of the logical connectives modelled by t-norms and related s-norms. In Sec.8.2 we analyze max-t fuzzy relation equations and related dual min-s fuzzy relation equations, assuming that t (resp s) is lower (resp. upper) semicontinuous. In Sec.8.3 we pay attention to equations of complex structure and a related adjoint fuzzy relation equation is studied in Sec.8.4. In the last Sec.8.5, max-t fuzzy relation equations under upper semicontinuous t-norms are studied. All the equations considered in this Ch. are assigned on finite referential sets.
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