Extended-order algebras and fuzzy implicators

In this work we reconsider the notion of implicator in a complete lattice L and discuss its properties, taking into account the viewpoint of the implication operation of classes of (weak) extended-order algebras, introduced by C. Guido and P. Toto and included in the class of implicative algebras considered by E. Rasiowa. In fact, such an implication, that is an extension of an order relation, can be viewed as an implicator in L, whose properties depend on those characterizing the structure of the algebra. We also propose in a (weak) right-distributive complete extended-order algebra (L,→,>) with adjoint product ⊗ a relative implication as a further implicator beyond →. The relative implication allows an extension of the inclusion relation between L-sets A and B, different from the subsethood degree, that consists in seeing to which extent A is included in its conjunction with B. Moreover, we introduce in (L,→,>) a further binary operation that we call conditional conjunction that can be read as ”a” and ”b, given a”, which motivates the term we have chosen to denote it. This operation, strictly related to the divisibility condition of BL algebras, satisfies most conditions usually asked to a conjunction and it is well behaved with the adjoint product ⊗ and the meet operation ∧.