ALGEBRAIC GENERATIONS OF SOME FUZZY POWERSET OPERATORS

In this paper, let L be a complete residuated lattice, and let Set denote the category of sets and mappings, LF-Pos denote the category of LF-posets and LF-monotone mappings, and LF-CSLat(?), LF-CSLat(?) 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 Set and LF- CSLat(?), between LF-Pos and LF-CSLat(?), and between LF-Pos and LF-CSLat(?), that is, Set? LF-CSLat(?), LF-Pos? LF-CSLat(?), and LF-Pos? LF-CSLat(?). And a usual mapping f generates the traditional Zadeh forward powerset operator f?l and the fuzzy forward powerset operators f?, ?f?* , ?f*? 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.