Monotone \(\varOmega \) -Sup-Fuzzy Relations: Converse and Complementation

L-fuzzy relations on a set X are functions from \(X \times X\) to the lattice L and act on the L-fuzzy subsets of X. When L is the lattice of sup-preserving endomaps on a complete lattice \(\varOmega \) , the relations act also on the \(\varOmega \) -fuzzy subsets of X. We call these relations equipped with this action, \(\varOmega \) -sup-fuzzy relations. When X is a preorder, monotone relations of this form act on the lattice of monotone functions from X to \(\varOmega \) . The motivation comes from mathematical morphology in image processing. Grey-scale images are modelled as functions on sets of pixels with \(\varOmega \) as the set of grey levels. More generally, graphs and hypergraphs labelled by grey levels can be handled. Enriching the lattice of \(\varOmega \) -sup-fuzzy relations with a multiplication operation provides a unital quantale that acts on the lattice of grey-scale images via the morphological operations of dilation and erosion. We study the quantale of \(\Omega \) -sup-fuzzy relations, with particular attention to the concepts of converse and complementation for these relations.