Hypergraph functor and attachment

Since the inception of the notion of fuzzy topology much attention was paid to the possible means of interaction between fuzzy and crisp topological settings, in order to see whether fuzzy topology was doing anything new. In particular, different functors relating the categories Top of topological spaces and L-Top of Ltopological spaces (L being a suitable complete lattice) appeared in the literature, providing the desired machinery for comparing classical and fuzzy developments. One of the mo st important examples in the field is hypergraph functor. Initiated by R. Lowen [8] and E. S. Santos [13], the concept was studied by many researchers [4],[6],[7],[11], but still failed to gai n much prominence in the fuzzy community. The main reasons were, firstly, the lack of information on functorial properties of the hypergraph functor and, secondly, remarkable differences in its definition by various authors (cf., e.g., those of U. Höhl e [6] and S. E. Rodabaugh [11]). There has been several attempts to amend the situation. The former of the above-mentioned deficiencies was partly removed by W. Kotzé, T. Kubiak [7] and U. Höhle [6] by considering the hypergraph functor from the categorical poin t of view. The second deficiency, however, appeared more resistant and was approached to only recently by C. Guido [5]. Motivated by the concept of quasicoincidence [9] (which is an analogue of the intersection property for fuzzy sets), he introduced the notion of attachment on a complete lattice as follows.