Fuzzy adjunctions and fuzzy morphological operations based on implications

The concept of adjunction plays an important role in mathematical morphology. If the morphological operations, dilation and erosion form an adjunction in a complete lattice, then they, as well as the closing and opening constructed by them, will fulfill certain required properties in an algebraic context. In the context of fuzzy mathematical morphology, which is an extension of binary morphology to gray-scale morphology based on fuzzy set theory, we use conjunctions and implications to define fuzzy dilations and fuzzy erosions. In this paper, we investigate when these pairs of dilations and erosions form a fuzzy adjunction, which is also defined by an implication. We find that the so-called adjointness between a conjunction and an implication plays an important role here. Finally, we develop a theorem stating that a conjunction that is adjoint with an implication cannot only be generated by an R-implication but also by other implications. This allows the easy construction of fuzzy adjunctions.