ALGEBRAIC GENERATIONS OF SOME FUZZY POWERSET OPERATORS

In this paper, let L be a completeresiduated lattice, and let {\bf Set} denote the category of setsand mappings, LF-{\bf Pos} denote the category of LF-posets andLF-monotone mappings, and LF-{\bf CSLat}(\sqcup), LF-{\bfCSLat}(\sqcap) denote the category of LF-completelattices and LF-join-preserving mappings and the category ofLF-complete lattices and LF-meet-preserving mappings, respectively. It isproved that there are adjunctions between {\bf Set} and LF-{\bf CSLat}(\sqcup), between LF-{\bfPos} and LF-{\bf CSLat}(\sqcup), and between LF-{\bf Pos} andLF-{\bf CSLat}(\sqcap), that is, {\bf Set}\dashv LF-{\bf CSLat}(\sqcup), LF-{\bfPos}\dashv LF-{\bf CSLat}(\sqcup), and LF-{\bf Pos}\dashvLF-{\bf CSLat}(\sqcap). And a usual mapping f generates thetraditional Zadeh forward powerset operator f_L^\rightarrow andthe 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 shownthat all the fuzzy powerset operators mentioned above can be generated by the underlying algebraic theories.

Keywords:

Complete residuated lattice , L-fuzzy poset , category , Adjunction , Algebraic theory , Powerset theory , Algebraic generation