Tang, Jian-Gang; Luo, Mao-Kang; Liu, Miao Free object in the category of \(L\)-fuzzy left \(R\)-modules determined by \(L\)-fuzzy set. (English) Zbl 1197.93043 Kybernetes 38, No. 3-4, 506-512 (2009). 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 Keywords:cybernetics; fuzzy logic; systems theory PDFBibTeX XMLCite \textit{J.-G. Tang} et al., Kybernetes 38, No. 3--4, 506--512 (2009; Zbl 1197.93043) Full Text: DOI References: [1] DOI: 10.1090/S0002-9904-1969-12267-6 · Zbl 0177.02401 · doi:10.1090/S0002-9904-1969-12267-6 [2] DOI: 10.1016/0022-247X(77)90233-5 · Zbl 0358.46011 · doi:10.1016/0022-247X(77)90233-5 [3] DOI: 10.1016/0893-9659(89)90087-6 · Zbl 0711.93001 · doi:10.1016/0893-9659(89)90087-6 [4] DOI: 10.1080/01969729108902311 · Zbl 0753.54004 · doi:10.1080/01969729108902311 [5] DOI: 10.1016/0165-0114(80)90025-1 · Zbl 0439.52001 · doi:10.1016/0165-0114(80)90025-1 [6] DOI: 10.1016/0165-0114(87)90156-4 · Zbl 0616.16013 · doi:10.1016/0165-0114(87)90156-4 [7] DOI: 10.1016/0165-0114(89)90181-4 · Zbl 0664.46009 · doi:10.1016/0165-0114(89)90181-4 [8] Tang, J. (1989), ”Latticed-valued algebra (I)”, Journal of Ili Teachers College, Vol. 1, pp. 1-17 (China). [9] Tang, J. (1991a), ”Free LF modules”, Fuzzy Systems and Mathematics, Vol. 2, pp. 23-6 (China). · Zbl 1210.16049 [10] Tang, J. (1991b), ”Latticed-valued algebra (II)”, Journal of Ili Teachers College, Vol. 1, pp. 13-26 (China). [11] Tang, J. (1993), ”Latticed-valued algebra (III)”, Journal of Ili Teachers College, Vol. 1, pp. 1-4 (China). [12] Tang, J. (1994), ”Latticed-valued algebra (IV)”, Journal of Ili Teachers College, Vol. 1, pp. 1-5 (China). [13] Tang, J. (1995), ”Tensor product and tensor functor of the category of L-fuzzy modules”, Fuzzy Systems and Mathematics, Vol. 3, pp. 65-73 (China). · Zbl 1266.16062 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.