Meet-continuity on $L$-directed Complete Posets

Document Type: Research Paper

Authors

1 College of Mathematics and Econometrics, Hunan University, Chang- sha 410082, P.R. CHINA and College of Science, East China Institute of Technology, Fuzhou, JiangXi 344000, P.R. China

2 College of Mathematics and Econometrics, Hunan University, Chang- sha 410082, P.R. China

Abstract

In this paper, the definition of meet-continuity on $L$-directed
complete posets (for short, $L$-dcpos) is introduced. As a
generalization of meet-continuity on crisp dcpos, meet-continuity on
$L$-dcpos, based on the generalized Scott topology, is
characterized. In particular, it is shown that every continuous
$L$-dcpo is meet-continuous and $L$-continuous retracts of
meet-continuous $L$-dcpos are also meet-continuous. Then, some
topological properties of meet-continuity on $L$-dcpos are
discussed. It is shown that meet-continuity on $L$-dcpos is a
topological invariant with respect to the generalized Scott
topology, and meet-continuity on $L$-dcpos is hereditary with
respect to generalized Scott closed subsets.

Keywords


bibitem{Fan:Naqdt} L. Fan, {it A new approach to quantitative domain theory}, Electronic Notes in Theoretical Computer Science, {bf 45} (2001), 77--87.
bibitem{Fan:Rspdt} L. Fan,
{it Researches on some problems in domain theory}, Ph. D. Thesis of
Capital Normal University, 2001.
bibitem{FaZhXiZh:LaqdIgosvfat} 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 Math (The Special
Issue of Theory of Fuzzy Sets and Application), {bf 14} (2000),
6--7.
bibitem{FlSuWa:Laqdt} B. Flagg, P. S"{u}nderhauf and K. Wagner, {it A logical approach to
quantitative domain theory}, Topology Atlas Preprint,
textbf{23} (1996), 10--29.
bibitem{GiHoKeLaMiSc:Cld} G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, {it Continuous lattices and
domains}, Cambridge University Press, 2003.

bibitem{KoLiLu:Omcd} H. Kou, Y. M. Liu and M. K. Luo, {it On meet-continuous
dcpos}, G. Q. Zhang et al. eds., Domain Theory, Logic and
Computation, Kluwer Academic Publishers, Netherlands, (2003),
117--135.


bibitem{LaZh:Cdcc} H. L. Lai and D. X. Zhang, {it Complete and directed complete
$Omega$-categories}, Theoretical Computer Science, {bf 388} (2007),
1--24.

bibitem{Rod:Pobffpsltfstt}S. E. Rodabaugh, {it Powerset operator based foundation for point-set lattice-theoretic
(poslat) fuzzy set theories and topologies}, Quaestiones
Mathematicae, {bf 20} (1997), 463--530.
bibitem{Rod:Poffpfstt}S. E. Rodabaugh, {it Powerset operator foundations for poslat fuzzy set theories and topologies, mathematics
of fuzzy sets: logic, topology, and measure theory}, (U. H"{o}hle
and S. E. Rodabaugh, eds.), The Hand books of Fuzzy Sets Series,
Dordrecht: Kluwer Academic Publishers, {bf 3} (1999), 91--116.
bibitem{Rod:Roatptfttflvm}S. E. Rodabaugh, {it Relationship of algebraic theories to
powerset theories and fuzzy topological theories for lattice-valued
mathematics}, International Journal of Mathematics and Mathematical
Sciences, {bf 2007} (2007), 1--71.

bibitem{Rut:Eogudt}J. J. M. M. Rutten, {it Elements of generalized ultrametric domain theory},
Theoretical Computer Science, {bf 170} (1996), 349--381.
bibitem{Wag:Srdewec} K. R. Wagner, {it Solving recursive
domain equations with enriched categories}, Ph.D Thesis, School of
Computer Science, Carnegie-Mellon Univerity, Pittsbrugh, 1994.
bibitem{Yao:Lfstscslfd} W. Yao, {it $L$-fuzzy
Scott topology and Scott convergence of stratified $L$-filters on
fuzzy dcpos}, Electronic Notes in Theoretical Computer Science, {bf
257} (2009), 135--152.
bibitem{Yao:QdfspIcfdcp} 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{Yao:Atfffp} W. Yao, {it An approach to fuzzy frames via fuzzy posets}, Fuzzy Sets and Systems, {bf 166} (2011), 75--89.

bibitem{Yao:Soffppias} W. Yao, {it A survey of fuzzifications of frames, the Papert-Papert-Isbell adjunction and
sobriety}, Fuzzy Sets and Systems, {bf 190} (2012),
63--81.
bibitem{YaSh:QdfspIIfstfdcp} W. Yao and F. G. Shi, {it
Quantitative domains via fuzzy sets: part II: fuzzy scott topology
on fuzzy directed complete posets}, Fuzzy Sets and Systems, {bf
173} (2011), 60--80.
bibitem{ZhXi:Agofpo} Q. Y. Zhang, {it Algebraic generations of
some fuzzy powerset operator}, Iranian Journal of Fuzzy Systems,
{bf 8} (2011), 31--58.
bibitem{ZhFa:Ciqd} Q. Y. Zhang and L. Fan,
{it Continuity in quantitative domains}, Fuzzy Sets and Systems,
{bf 154} (2005), 118--131.
bibitem{ZhXi:Gstolfd} Q. Y. Zhang and W. X. Xie, {it The
generalized Scott topologies on $L$-fuzzzy domains}, Journal of
Mathematics, (Chinese), {bf 26} (2006), 312--318.
bibitem{ZhXi:Srpbfd} Q. Y. Zhang and W. X. Xie, {it Section-retraction-pairs between fuzzy domains}, Fuzzy
Sets and Systems, {bf 158} (2007), 99--114.
bibitem{ZhXiFa:Fcl} Q. Y. Zhang, W. X. Xie and L. Fan, {it Fuzzy complete lattices}, Fuzzy
Sets and Systems, {bf 160} (2009), 2275--2291.

bibitem{ZhZh:Mvltr} H. B. Zhao and D. X. Zhang, {it
Many valued lattices and their representations}, Fuzzy Sets and
Systems, {bf 159} (2008), 81--94.