On D-implications derived by grouping functions

It is common sense the relevance of overlap and grouping functions, specially in applications in which associativity is not required. In this work, based on previous investigations concerning classes of implication functions derived from overlap functions O, grouping functions G and fuzzy negations N (namely, (G, N)-implications, R O -implications and QL-implications constructed from (O, G, N)), a deep study on D-implications constructed from grouping functions G is carried out. Such fuzzy implications, also known as Dishkant implications, are obtained from D-operations derived from (O, G, N), which are used for the generalization of the implication p → q ≡ q∨(¬p∧¬q) of orthomodular lattices. We investigate under which conditions such D-operations are fuzzy implication functions, presenting a general form for obtaining D-implications. We also provide a comparative study of D-implications derived from grouping functions and other classes of fuzzy implications constructed from N, O and G, analysing the intersections among such classes.