Document Type : Research Paper

Author

• Qi-Ye Zhang

School of Mathematics and Systems Science, Beihang University, Beijing 100191, China and LMIB of the Ministry of Education, Beijing 100191, China

Abstract

In this paper, let $L$ be a complete
residuated lattice, and let {\bf Set} denote the category of sets
and mappings, $LF$-{\bf Pos} denote the category of $LF$-posets and
$LF$-monotone mappings, and $LF$-{\bf CSLat}$(\sqcup)$, $LF$-{\bf
CSLat}$(\sqcap)$ 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 {\bf Set} and $LF$-{\bf CSLat}$(\sqcup)$, between $LF$-{\bf
Pos} and $LF$-{\bf CSLat}$(\sqcup)$, and between $LF$-{\bf Pos} and
$LF$-{\bf CSLat}$(\sqcap)$, that is, {\bf Set}$\dashv LF$-{\bf CSLat}$(\sqcup)$, $LF$-{\bf
Pos}$\dashv LF$-{\bf CSLat}$(\sqcup)$, and $LF$-{\bf Pos}$\dashv$
$LF$-{\bf CSLat}$(\sqcap)$. And a usual mapping $f$ generates the
traditional Zadeh forward powerset operator $f_L^\rightarrow$ and
the 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 shown
that 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