$L-$ordered Fuzzifying Convergence Spaces

Document Type: Research Paper

Authors

Department of Mathematics, Ocean University of China, 266100 Qing- dao, P. R. China

Abstract

Based on a complete Heyting algebra, we modify the definition of
lattice-valued fuzzifying convergence space using fuzzy inclusion
order and construct in this way a Cartesian-closed category, called
the category of $L-$ordered fuzzifying convergence spaces, in which
the category of $L-$fuzzifying topological spaces can be embedded.
In addition, two new categories are introduced, which are called the
category of principal $L-$ordered fuzzifying convergence spaces and
that of topological $L-$ordered fuzzifying convergence spaces, and
it is shown that they are isomorphic to the category of
$L-$fuzzifying neighborhood spaces and that of $L-$fuzzifying
topological spaces respectively.

Keywords


bibitem{1} H. Boustique, R. N. Mohapatra and G. Richardson, {it Lattice-valued fuzzy interior operators}, Fuzzy Sets and Systems, {bf 160} (2009), 2947-2955.
bibitem{2} H. Boustique and G. Richardson, {it A note on regularity}, Fuzzy Sets and Systems, {bf 162} (2011), 64-66.
bibitem{3} J. Fang, {it Stratified $L-$ordered convergence structures}, Fuzzy Sets and Systems, {bf 161} (2010), 2130-2149.
bibitem{4} P. V. Flores, R. N. Mohapatra and G. Richardson, {it Lattice-valued spaces: fuzzy convergence}, Fuzzy Sets and Systems, {bf 157} (2006), 2706-2714.
bibitem{5} P. V. Flores and G. Richardson, {it  Lattice-valued convergence: diagonal axioms}, Fuzzy Sets and Systems, {bf 159} (2008), 2520-2528.
bibitem{6}  U. H"{o}hle, {it Characterization of $L-$topologies by $L-$valued neighborhoods}, Chapter 5, In:
Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The
Handbooks of Fuzzy Sets Series, (U. H"{o}hle, S. E. Rodabaugh,
eds.), Kluwer Academic Publishers, Boston, Dordrecht,
London, {bf3} (1999), 389-432.
bibitem{7}  U. H"{o}hle and A. P. v{S}ostak, {it Axiomatic foundations of fixed-basis fuzzy topology},
In: Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory,
The Handbooks of Fuzzy Sets Series, (U. H"{o}hle, S. E. Rodabaugh,
eds.), Kluwer Academic Publishers, Boston, Dordrecht,
London, {bf3} (1999), 123-173.
bibitem{8} G. J"{a}ger, {it A category of $L-$fuzzy convergence spaces}, Quaestiones Mathematicae, {bf 24} (2001), 501-517.
bibitem{9} G. J"{a}ger, {it Subcategories of lattice-valued convergence spaces}, Fuzzy Sets and Systems, {bf 156} (2005), 1-24.
bibitem{10} G. J"{a}ger, {it Pretopological and topological lattice-valued convergence spaces}, Fuzzy Sets and Systems, {bf 158} (2007), 424-435.
bibitem{11} G. J"{a}ger, {it Fischer's diagonal condition for lattice-valued convergence spaces}, Quaestiones Mathematicae, {bf 31} (2008), 11-25.
bibitem{12} R. Lowen, {it Convergence in fuzzy topological spaces}, Gen. Top. Appl., {bf 10} (1979), 147-160.
bibitem{13} K. C. Min, {it Fuzzy limit spaces}, Fuzzy Sets and Systems, {bf 32} (1989), 343-357.
bibitem{14} L. Xu, {it Characterizations of fuzzifying topologies by some limit structures}, Fuzzy Sets and Systems, {bf 123} (2001), 169-176.
bibitem{15} W. Yao, {it On many-valued stratified $L-$fuzzy convergence spaces}, Fuzzy Sets and Systems, {bf 159} (2008), 2503-2519.
bibitem{16} W. Yao, {it On $L-$fuzzifying convergence spaces}, Iranian Journal of Fuzzy Systems, {bf 6}textbf{(1)} (2009), 63-80.
bibitem{17} M. S. Ying, {it A new approach to fuzzy topology (I)}, Fuzzy Sets and Systems, {bf 39} (1991), 303-321.
bibitem{18} D. Zhang, {it On the reflectivity and coreflectivity of $L-$fuzzifying topological spaces in $L-$topological
spaces}, Acta Mathematica Sinica (English Series), {bf
18}textbf{(1)} (2002), 55-68.
bibitem{19} D. Zhang, {it $L-$fuzzifying topologies as $L-$topologies}, Fuzzy Sets and Systems, {bf 125} (2002), 135-144.
bibitem{20} D. Zhang and L. Xu, {it Categories isomorphic to bf FNS}, Fuzzy Sets and Systems, {bf 104} (1999), 373-380.