F-transforms determined by overlap and grouping maps over a complete lattice

This work examines F-transforms based on overlap and grouping maps, residual, and co-residual implicator over complete lattice from constructive and axiomatic approaches, wherein the results highly depend upon the negator. Specifically, (i) we initiate the study of F-transforms using the concepts of the overlap and grouping maps, which can be viewed as a new structure of classical logic’s intersection and union operations on the unit interval [0, 1], (ii) we show that the proposed F-transforms generalize the existing such transforms, and (iii) some of the results do not hold for general implicators. Further, we study the concept of fuzzy transformation systems and examine the duality.