Free object in the category of L-fuzzy left R-modules determined by L-fuzzy set
Purpose - The purpose of this paper is to study free L-fuzzy left R-module, using the language of categories and functors for the general description of L-fuzzy left R-modules generated by L-fuzzy set. In the language of categories and functors, an L-fuzzy left R-modules generated by L-fuzzy set is called a free object in the category of L-fuzzy left R-modules determined by L-fuzzy set. Design/methodology/approach - Category theory is used to study the existent quality, unique quality and material structure of L-fuzzy left R-modules generated by L-fuzzy set. Findings - The paper gives the uniqueness, structure and existence theorems of free object in the category of L-fuzzy left R-modules determined by L-fuzzy set, and the authors prove that the fuzzy free functor is left adjoint to the fuzzy underlying functor. Research limitations/implications - Some property of free L-fuzzy left R-modules will need to be further researched. Originality/value - The paper defines a new class of L-fuzzy left R-modules, i.e. free L-fuzzy left R-modules, research and explore free L-fuzzy left R-modules in theory.
fuzzy logic cybernetics systems theory