Free object in the category of L‐fuzzy left R‐modules determined by L‐fuzzy set

Abstract

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.

Citation

Tang, J., Luo, M. and Liu, M. (2009), "Free object in the category of L‐fuzzy left R‐modules determined by L‐fuzzy set", Kybernetes, Vol. 38 No. 3/4, pp. 506-512. https://doi.org/10.1108/03684920910944227