^{}School of Mathematics and Systems Science, Beihang University, Beijing
100191, China and LMIB of the Ministry of Education, Beijing 100191, China

Abstract

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.

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