# ALGEBRAIC GENERATIONS OF SOME FUZZY POWERSET OPERATORS

Document Type: Research Paper

Author

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

### References

bibitem{} L. Fan, Q. Y. Zhang, W. Y. Xiang and C. Y. Zheng,
{it An $L$-fuzzy approach to quantitative domain(I)-generalized
ordered set valued in frame and adjunction theory}, Fuzzy Systems and
Mathematics (The Special Issue of Theory of Fuzzy Sets and Application), In Chinese,
{bf 14} (2000), 6-7.
bibitem{} L. Fan, {it Research of some problems in
domain theory}, Ph.D. Thesis of Capital Normal University, Beijing, In
Chinese, 2001.

bibitem{} L. Fan, {it A new approach to quantitative domain theory}, Electronic Notes
in Theoretic Computer Science, http://www.elsevier.nl/locate/entcs, {bf 45} (2001), 77-87.

bibitem{} J. A. Goguen, {it $L$-Fuzzy sets}, Journal of Mathematical Analysis and Application, {bf 18} (1967), 145-174.

bibitem{} U. H"{o}hle and S. E. Rodabaugh, eds., {it Mathematics of fuzzy sets: logic, topology, and
measure theory, The Handbooks of Fuzzy Sets Series},
Kluwer Academic Pubers (Boston/Dordrecht/London), {bf 3} (1999).

bibitem{} G. M. Kelly, {it Basic concepts of enriched category theory}, London Mathematical Soceity Lecture Notes Series {bf 64}, Cambridge University Press, 1982. Also: Reprints in Theory and Applications of Categories,
{bf 10} (2005).

bibitem{} H. L. Lai and D. X. Zhang, {it Complete and directed complete
$Omega$-categories}, Theoretical Computer Science, {bf 388} (2007), 1-25.

bibitem{} S. Mac Lane, {it Categories for the working mathematician (2nd
edition)}, Springer-Verlag (Berlin/Heidelberg/New York), 2003.

bibitem{} E. G. Manes, {it Algebraic theories}, Springer Verlag (Berlin/Heidelberg/New York), 1976.

bibitem{} S. E. Rodabaugh, {it Point-set lattice-theoretic topology}, Fuzzy Sets and Systems, {bf 40(2)} (1991), 297-345 .

bibitem{} S. E. Rodabaugh, {it Powerset operator based foundation for point-set lattice-theoretic (POSLAT) fuzzy set theories and topologies}, Quaestiones Mathematicae, {bf 20(3)} (1997), 463-530.

bibitem{} S. E. Rodabaugh, {it Powerset operator foundations for poslat fuzzy set theories and topologies}, Chapter 2 in [5], 91-116.

bibitem{} S. E. Rodabaugh, {it Relationship of algebraic theories to powerset theories and fuzzy topological
theories for lattice-valued mathematics}, International Journal of
Mathematics and the Mathematical Sciences {bf 3}, Article ID
43645, doi:10.1155/2007/43645, (2007), 71.

bibitem{} K. R. Wagner, {it Solving recursive domain equations with enriched categories}, Ph. D. Thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, 1994.

bibitem{} W. Yao and L. X. Lu, {it Fuzzy Galois connections on fuzzy posets}, Mathematical Logic Quarterly, {bf 55} (2009), 105-112.

bibitem{} W. Yao, {it Quantitative domains via fuzzy sets: part I: continuity of fuzzy directed complete posets}, Fuzzy Sets and Systems, {bf 161} (2010), 973-987.

bibitem{} L. A. Zadeh, {it Fuzzy Sets}, Information and Control, {bf 8} (1965), 338-353.
bibitem{} Q. Y. Zhang and L. Fan, {it Continuity in quantitative domains}, Fuzzy Sets and Systems, {bf 154} (2005), 118-131.
bibitem{} Q. Y. Zhang and L. Fan, {it A kind of $L$-fuzzy complete lattices
and adjoint functor theorem for $LF$-posets}, Report on the Fourth
International Symposium on Domain Theory, Hunan University,
Changsha, China, June 2006.
bibitem{} Q. Y. Zhang and W. X. Xie, {it Fuzzy complete lattices}, Fuzzy Sets and Systems, {bf 160} (2009), 2275-2291.
bibitem{} Q. Y. Zhang, L. Fan and W. X. Xie, {it Adjoint functor theorem for fuzzy posets}, Indian Journal of Mathematics, {bf 51} (2009), 305-342.