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Free object in the category of \(L\)-fuzzy left \(R\)-modules determined by \(L\)-fuzzy set. (English) Zbl 1197.93043

Summary: 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.
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.
The paper gives the uniqueness, structure and existence theorems of a 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.
Some property of free \(L\)-fuzzy left \(R\)-modules need to be further researched.
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.

MSC:

93A10 General systems
93C42 Fuzzy control/observation systems
16Y99 Generalizations
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