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representation theorems of $L-$subsets and $L-$families on complete residuated lattice | ||
| Iranian Journal of Fuzzy Systems | ||
| مقاله 9، دوره 10، شماره 3، شهریور 2013، صفحه 125-136 اصل مقاله (398.13 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22111/ijfs.2013.866 | ||
| نویسندگان | ||
| Hui Han* 1؛ Jinming Fang2 | ||
| 1Department of Mathematics, Ocean University of China, 266100 Qingdao, P.R. China | ||
| 2Department of Mathematics, Ocean University of China, 266100 Qing- dao, P.R. China | ||
| چکیده | ||
| In this paper, our purpose is twofold. Firstly, the tensor and residuum operations on $L-$nested systems are introduced under the condition of complete residuated lattice. Then we show that $L-$nested systems form a complete residuated lattice, which is precisely the classical isomorphic object of complete residuated power set lattice. Thus the new representation theorem of $L-$subsets on complete residuated lattice is obtained. Secondly, we introduce the concepts of $L-$family and the system of $L-$subsets, then with the tool of the system of $L-$subsets, we obtain the representation theorem of intersection-preserving $L-$families on complete residuated lattice. | ||
| کلیدواژهها | ||
| Complete residuated lattices؛ $L-$subsets؛ $L-$nested systems؛ $L-$families؛ Level $L-$subsets؛ Representation theorems | ||
| مراجع | ||
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[1] R. Belohlavek, Fuzzy relational systems: foundations and principles, Kluwer Academic/ Plenum Press, New York, 2002. [2] J. M. Fang and Y. L. Han, A new representation theorem of L-Sets, Periodical of Ocean University of China (Natural Science), 38(6) (2008), 1025-1028. [3] M. Gorjanac-Ranitovic and A. Tepavcevic, General form of lattice-valued fuzzy sets under the cutworthy approach, Fuzzy Sets and Systems, 158(11) (2007), 1213-1216. [4] G. Jager, Level spaces for lattice-valued uniform convergence spaces, Quaest. Math, 31 (2008), 255-277. [5] C. Z. Luo, Fuzzy sets and nested systems, Fuzzy Mathematics, 3(4) (1983), 113-126. [6] C. V. Negoita and D. A. Ralescu, Representation theorems for fuzzy concepts, Kybernetes, 4 (1975), 169-174. [7] B. Seselja and A. Tepavcevic, A note on a natural equivalence relation on fuzzy power set, Fuzzy Sets and Systems, 148(2) (2004), 201-210. [8] B. Seselja and A. Tepavcevic, Representing ordered structures by fuzzy sets: an overview, Fuzzy Sets and Systems, 136(1) (2003), 21-39. [9] B. Seselja and A. Tepavcevic, Completion of ordered structures by cuts of fuzzy sets: an overview, Fuzzy Sets and Systems, 136(1) (2003), 1-19. [10] F. G. Shi, Theory of L -nested sets and L -nested sets and applications, Fuzzy Systems and Mathematics, 9(4) (1995), 65-72. [11] M. Voskoglou, Measuring students modeling capacities: a fuzzy approach, Iranian Journal of Fuzzy Systems, 8(3) (2011), 23-33. [12] F. L. Xiong, The representation theorems on complete lattice and their application, Periodical of Ocean University of Qingdao, 28(2) (1998), 339-344. [13] W. X. Zhang, Fuzzy mathematic basis, Xi'an: Xi'an Jiao Tong University Press, 1984. [14] Q. Y. Zhang, Algebraic generations of some fuzzy powerset operators, Iranian Journal of Fuzzy Systems, 8(5) (2011), 31-58. | ||
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