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 LP-monotone mappings, and LF-CSLat(boolean OR), LF-CSLat(boolean AND) denote the category of LP-complete lattices and LP-join-preserving mappings and the category of LP-complete lattices and LF-meet-preserving mappings, respectively. It is proved that there are adjunetions between Set and LF-CSLat(boolean OR), between LF-Pos and LF-CSLat(boolean OR), and between LF-Pos and LF-CSLat(boolean AND), that is, Set-I LF-CSLat(boolean OR), LF-Pos-I LP-CSLat(boolean OR), and LF-Pos-1 LP-CSLat(boolean AND). And a usual inapping f generates the traditional Zadeh forward powerset operator f(L)(->) and the fuzzy forward powerset operators (f) over tilde (->) (f) over tilde.(->) (f) over tilde*(->) defined by the author at 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.