Coreflectivities of (L, M)-fuzzy convex structures and (L, M)-fuzzy cotopologies in (L, M)-fuzzy closure systems

In this paper, we mainly focus on the relationship between (L, M)-fuzzy closure systems and (L, M)-fuzzy convex structures as well as the relationship between (L, M)-fuzzy closure systems and (L, M)-fuzzy cotopologies. Firstly, we show that there is an adjunction between the category LMFC of (L, M)-fuzzy convex spaces and the category LMFS of (L, M)-fuzzy closure spaces, and there is also an adjunction between the category LMFT of (L, M)-fuzzy cotopological spaces and LMFS. In particular, the categories LMFC and LMFT are both coreflective subcategories of LMFS. Secondly, we prove that there is an adjunction between the category ELFC of extensional L-fuzzy convex spaces and the category LFC of L-fuzzy convex spaces, and there is also an adjunction between the category ELFT of extensional L-fuzzy cotopological spaces and the category LFT of (L, M)-fuzzy cotopological spaces. Specially, ELFC is a coreflective subcategory of LFC and ELFT is a coreflective subcategory of LFT.