Fuzzifying Modal Algebra

Fuzzy relations are mappings from pairs of elements into the interval [0,1]. As a replacement for the complement operation one can use the mapping that sends x to 1 − x. Together with the concepts of t-norm and t-conorm a weak form of Boolean algebra can be defined. However, to our knowledge so far no notion of domain or codomain has been investigated for fuzzy relations. These might, however, be useful, since fuzzy relations can, e.g., be used to model flow problems and many other things. We give a new axiomatisation of two variants of domain and codomain in the more general setting of idempotent left semirings that avoids complementation and hence is applicable to fuzzy relations. Some applications are sketched as well.