Gradual and Fuzzy Modules: Functor Categories

Abstract

The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod−P, which is a Grothendieck category. To do that, first we consider the preadditive category P, defined by the interval P=(0,1], to build a torsionfree class J in Mod−P, and a hereditary torsion theory in Mod−P, to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G,F), of decreasing gradual submodules of M, where G belongs to J, satisfying G=Fd, and ∪αF(α) is a disjoint union of F(1) and F(α)\G(α), where α is running in (0,1].

Gradual Sets: An Approach to Fuzzy Sets

“…Tis situation allows us to formulate a more attractive category theory of gradual objects which includes the usual constructions of the category of groups. In this context, decreasing gradual groups, strict decreasing gradual groups and fuzzy groups can be identifed with adequate subcategories; see the forthcoming paper [10], in which we study gradual and fuzzy modules over a ring.…”