اخبار و اعلانات
آمار
| تعداد نشریات | 27 |
| تعداد شماره‌ها | 604 |
| تعداد مقالات | 6,159 |
| تعداد مشاهده مقاله | 9,098,545 |
| تعداد دریافت فایل اصل مقاله | 5,935,433 |
ALGEBRAIC GENERATIONS OF SOME FUZZY POWERSET OPERATORS | ||
| Iranian Journal of Fuzzy Systems | ||
| مقاله 4، دوره 8، شماره 5، دی 2011، صفحه 31-58 اصل مقاله (331.38 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22111/ijfs.2011.296 | ||
| نویسنده | ||
Qi-Ye Zhang 
| ||
| School of Mathematics and Systems Science, Beihang University, Beijing 100191, China and LMIB of the Ministry of Education, Beijing 100191, China | ||
| چکیده | ||
| In this paper, let $L$ be a complete residuated lattice, and let {\bf Set} denote the category of sets and mappings, $LF$-{\bf Pos} denote the category of $LF$-posets and $LF$-monotone mappings, and $LF$-{\bf CSLat}$(\sqcup)$, $LF$-{\bf CSLat}$(\sqcap)$ denote the category of $LF$-complete lattices and $LF$-join-preserving mappings and the category of $LF$-complete lattices and $LF$-meet-preserving mappings, respectively. It is proved that there are adjunctions between {\bf Set} and $LF$-{\bf CSLat}$(\sqcup)$, between $LF$-{\bf Pos} and $LF$-{\bf CSLat}$(\sqcup)$, and between $LF$-{\bf Pos} and $LF$-{\bf CSLat}$(\sqcap)$, that is, {\bf Set}$\dashv LF$-{\bf CSLat}$(\sqcup)$, $LF$-{\bf Pos}$\dashv LF$-{\bf CSLat}$(\sqcup)$, and $LF$-{\bf Pos}$\dashv$ $LF$-{\bf CSLat}$(\sqcap)$. And a usual mapping $f$ generates the traditional Zadeh forward powerset operator $f_L^\rightarrow$ and the fuzzy forward powerset operators $\widetilde{f}^\rightarrow, \widetilde{f}_\ast^\rightarrow, \widetilde{f}^{\ast\rightarrow}$ defined by the author et al via these adjunctions. Moreover, it is also shown that all the fuzzy powerset operators mentioned above can be generated by the underlying algebraic theories. | ||
| کلیدواژه‌ها | ||
| Complete residuated lattice؛ $L$-fuzzy poset؛ category؛ Adjunction؛ Algebraic theory؛ Powerset theory؛ Algebraic generation | ||
| مراجع | ||
|
\bibitem{} L. Fan, Q. Y. Zhang, W. Y. Xiang and C. Y. Zheng, {\it An $L$-fuzzy approach to quantitative domain(I)-generalized ordered set valued in frame and adjunction theory}, Fuzzy Systems and Mathematics (The Special Issue of Theory of Fuzzy Sets and Application), In Chinese, {\bf 14} (2000), 6-7. \bibitem{} L. Fan, {\it Research of some problems in domain theory}, Ph.D. Thesis of Capital Normal University, Beijing, In Chinese, 2001. \bibitem{} L. Fan, {\it A new approach to quantitative domain theory}, Electronic Notes in Theoretic Computer Science, http://www.elsevier.nl/locate/entcs, {\bf 45} (2001), 77-87. \bibitem{} J. A. Goguen, {\it $L$-Fuzzy sets}, Journal of Mathematical Analysis and Application, {\bf 18} (1967), 145-174.
\bibitem{} U. H\"{o}hle and S. E. Rodabaugh, eds., {\it Mathematics of fuzzy sets: logic, topology, and measure theory, The Handbooks of Fuzzy Sets Series}, Kluwer Academic Pubers (Boston/Dordrecht/London), {\bf 3} (1999). \bibitem{} G. M. Kelly, {\it Basic concepts of enriched category theory}, London Mathematical Soceity Lecture Notes Series {\bf 64}, Cambridge University Press, 1982. Also: Reprints in Theory and Applications of Categories, {\bf 10} (2005). \bibitem{} H. L. Lai and D. X. Zhang, {\it Complete and directed complete $\Omega$-categories}, Theoretical Computer Science, {\bf 388} (2007), 1-25. \bibitem{} S. Mac Lane, {\it Categories for the working mathematician (2nd edition)}, Springer-Verlag (Berlin/Heidelberg/New York), 2003. \bibitem{} E. G. Manes, {\it Algebraic theories}, Springer Verlag (Berlin/Heidelberg/New York), 1976.
\bibitem{} S. E. Rodabaugh, {\it Point-set lattice-theoretic topology}, Fuzzy Sets and Systems, {\bf 40(2)} (1991), 297-345 .
\bibitem{} S. E. Rodabaugh, {\it Powerset operator based foundation for point-set lattice-theoretic (POSLAT) fuzzy set theories and topologies}, Quaestiones Mathematicae, {\bf 20(3)} (1997), 463-530.
\bibitem{} S. E. Rodabaugh, {\it Powerset operator foundations for poslat fuzzy set theories and topologies}, Chapter 2 in [5], 91-116.
\bibitem{} S. E. Rodabaugh, {\it Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics}, International Journal of Mathematics and the Mathematical Sciences {\bf 3}, Article ID 43645, doi:10.1155/2007/43645, (2007), 71. \bibitem{} K. R. Wagner, {\it Solving recursive domain equations with enriched categories}, Ph. D. Thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, 1994.
\bibitem{} W. Yao and L. X. Lu, {\it Fuzzy Galois connections on fuzzy posets}, Mathematical Logic Quarterly, {\bf 55} (2009), 105-112.
\bibitem{} W. Yao, {\it Quantitative domains via fuzzy sets: part I: continuity of fuzzy directed complete posets}, Fuzzy Sets and Systems, {\bf 161} (2010), 973-987.
\bibitem{} L. A. Zadeh, {\it Fuzzy Sets}, Information and Control, {\bf 8} (1965), 338-353. \bibitem{} Q. Y. Zhang and L. Fan, {\it Continuity in quantitative domains}, Fuzzy Sets and Systems, {\bf 154} (2005), 118-131. \bibitem{} Q. Y. Zhang and L. Fan, {\it A kind of $L$-fuzzy complete lattices and adjoint functor theorem for $LF$-posets}, Report on the Fourth International Symposium on Domain Theory, Hunan University, Changsha, China, June 2006. \bibitem{} Q. Y. Zhang and W. X. Xie, {\it Fuzzy complete lattices}, Fuzzy Sets and Systems, {\bf 160} (2009), 2275-2291. \bibitem{} Q. Y. Zhang, L. Fan and W. X. Xie, {\it Adjoint functor theorem for fuzzy posets}, Indian Journal of Mathematics, {\bf 51} (2009), 305-342. | ||
|
آمار تعداد مشاهده مقاله: 2,525 تعداد دریافت فایل اصل مقاله: 1,581 |
||
