Gradual and Fuzzy Modules: Functor Categories

Josefa M. García and Pascual Jara ()
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Josefa M. García: Department of Applied Mathematics, University of Granada, E-18071 Granada, Spain
Pascual Jara: Department of Algebra and IEMath, Instituto de Matemáticas, University of Granada, E-18071 Granada, Spain

Mathematics, 2022, vol. 10, issue 22, 1-21

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 = F d , and ∪ α F ( α ) is a disjoint union of F ( 1 ) and F ( α ) \ G ( α ) , where α is running in ( 0 , 1 ] .

Keywords: fuzzy set; fuzzy module; gradual element; gradual module; gradual ring; functorial category (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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