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 and$lf$-monotone mappings, and $lf$-{bf cslat}$(sqcup)$, $lf$-{bfcslat}$(sqcap)$ denote the category of $lf$-completelattices and $lf$-join-preserving mappings and the category of$lf$-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} and$lf$-{bf cslat}$(sqcap)$, that is, {bf set}$dashv lf$-{bf cslat}$(sqcup)$, $lf$-{bfpos}$dashv lf$-{bf cslat}$(sqcup)$, and $lf$-{bf pos}$dashv$$lf$-{bf cslat}$(sqcap)$. and a usual mapping $f$ generates thetraditional zadeh forward powerset operator $f_l^rightarrow$ andthe fuzzy forward powerset operators $widetilde{f}^rightarrow,widetilde{f}_ast^rightarrow, widetilde{f}^{astrightarrow}$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.