Qi-Ye Zhang - School of Mathematics and Systems Science, Beihang University, Beijing ۱۰۰۱۹۱, China and LMIB of the Ministry of Education, Beijing ۱۰۰۱۹۱, China

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.

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