Weak Converse and Complement for Quantale-Enriched Profunctors

Quantale-enriched profunctors are known to generalise binary relations between sets in two directions. Firstly, we have quantale-enriched categories instead of sets and, secondly, relatedness is quantale-valued rather than two-valued. For suitable quantales, there are natural generalisations of the converse and complement operations on profunctors. Weaker notions of converse and complement have already been studied, for example with monotone relations between preorders. However, these weaker operations for profunctors enriched over quantales have not been explored. In this paper, we focus on profunctors enriched over commutative and Girard quantales. We provide constructions for left and right converse and complement operations for these profunctors. In addition, we study the properties of these new operations and describe their applications to graph mathematical morphology.