Document Type : Research Paper


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


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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