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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 | ||
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