Monotone Ω-Sup-Fuzzy Relations: Converse and Complementation

L-fuzzy relations on a set X are functions from X x 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 Ω, the relations act also on the Ω-fuzzy subsets of X. We call these relations equipped with this action, Ω-sup-fuzzy relations. When X is a preorder, monotone relations of this form act on the lattice of monotone functions from X to Ω. The motivation comes from mathematical morphology in image processing. Grey-scale images are modelled as functions on sets of pixels with Ω as the set of grey levels. More generally, graphs and hypergraphs labelled by grey levels can be handled. Enriching the lattice of Ω-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 Ω-sup-fuzzy relations, with particular attention to the concepts of converse and complementation for these relations.