Fuzzy relations; Sup-T composition; Inf-R composition; Bandler-Kohout products

Fuzzy relational compositions have been extensively studied by many authors. Especially, we would like to express studies of the fuzzy relational compositions motivated by their applications to medical diagnosis by Willis Bandler and Ladislav Kohout. These types of compositions use only two quantifiers: universal quantifier and existential quantifier. There exists a huge gap between these quantifiers. This is a natural motivation for studying fuzzy relational compositions defined based on generalized quantifiers. In this paper, we revisit and provide definitions of these compositions as well as those of derived images and preimages. Furthermore, we investigate their mathematical properties.