Categories isomorphic to the category of $L$-fuzzy closure system spaces

Document Type : Research Paper

Authors

School of Mathematics, Beijing Institute of Technology, 5 South Zhong- guancun Street, Haidian District, 100081 Beijing, P.R. China

Abstract

In this paper, new definitions of $L$-fuzzy closure
 operator, $L$-fuzzy interior operator, $L$-fuzzy remote neighborhood
 system, $L$-fuzzy neighborhood system and $L$-fuzzy quasi-coincident neighborhood system
 are proposed. It is proved that the category of $L$-fuzzy closure spaces, the category of $L$-fuzzy interior spaces, the category of $L$-fuzzy remote neighborhood
 spaces, the category of $L$-fuzzy quasi-coincident neighborhood spaces, the category of $L$-fuzzy
 neighborhood spaces are all isomorphic to the category
 $L$-{\bf FCS} of $L$-fuzzy closure system spaces.

Keywords


\bibitem{bel} R. B\v{e}lohl\'{a}vek, {\it Fuzzy closure operators}, J. Math. Anal. Appl., {\bf 262} (2001), 473--489.

 \bibitem{bia1} L. Biacino and G. Gerla, {\it Closure systems and
  $L$-subalgebras}, Information Sciences, {\bf 33} (1984), 181--195.

 \bibitem{bia2} L. Biacino and G. Gerla, {\it An extension principle for
  closure operators}, J. Math. Anal. Appl., {\bf 198} (1996), 1--24.

 \bibitem{bir} G. Birkhoff, {\it Lattice theory}, 3rd Edition, Amer. Math.
  Soc., Rhode Island, 1967.

 \bibitem{cha} M. K. Chakraborty and J. Sen, {\it MV-algebras embedded in
  a CL-algebra}, Int. J. Approximate Reasoning, {\bf 18} (1998), 217--229.

  \bibitem{chat} K. C. Chattopadhyay and S. K. Samanta, {\it Fuzzy topology: fuzzy closure operator, fuzzy compactness and fuzzy connectedness}, Fuzzy Sets and Systems, {\bf 54} (1993), 207--212.

 \bibitem{fang1} J. M. Fang, {\it Categories isomorphic to $L$-\bf{FTOP}}, Fuzzy Sets and Systems, {\bf 157} (2006), 820--831.

 \bibitem{fang} J. M. Fang and Y. L. Yue, {\it $L$-fuzzy closure systems}, Fuzzy Sets and Systems, {\bf 161} (2010), 1242--1252.

 \bibitem{ger1} G. Gerla,  {\it An extension principle for fuzzy logics},
  Math. Logic Quart., {\bf 40} (1994), 357--380.

 \bibitem{ger2} G. Gerla, {\it Graded consequence relations and fuzzy
  closure operators}, J. Appl. Non-Classical Logics, {\bf 6} (1996), 369--379.


 \bibitem{gha}M. H. Ghanim, O. A. Tantawy and F. M. Selin, {\it Gradation of
  supra-openness}, Fuzzy Sets and Systems, {\bf 109} (2000), 245--250.

  \bibitem{goe} R. Goetschel and W. Voxman, {\it Spanning properties for fuzzy matroids}, Fuzzy Sets and Systems, {\bf 51} (1992), 313--321.

 \bibitem{hoh} U. H\"ohle and S. E. Rodabaugh, et al., (eds), {\it Mathematics of fuzzy sets: logic, topology and measure theory}, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, Boston, Dordrecht, London, {\bf 3} (1999).

 \bibitem{hoh2}U. H\"{o}hle and A. P. \v{S}ostak, {\it Axiomatic  foundations of fixed-basis fuzzy topology}, In \cite{hoh}, 123--173.

 \bibitem{kim1} Y. C. Kim, {\it Initial $L$-fuzzy closure spaces}, Fuzzy Sets and Systems, {\bf 133} (2003), 277--297.

 \bibitem{kim} Y. C. Kim and J. M. Ko, {\it Fuzzy closure systems and fuzzy
 closure operators}, Commun. Korean Math. Soc., {\textbf 19}{\bf(1)} (2004), 35--51.

 \bibitem{Kubiak} T. Kubiak, {\it On fuzzy topologies}, Ph. D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.

 \bibitem{kub} T. Kubiak and A. P. \v{S}ostak, {\it Foundations of the theory of $(L,M)$-fuzzy topological spaces}, Abstracts of the 30th Linz Seminar on Fuzzy Set Theory (U. Bodenhofer, B. De Bates, E. P. Klement, and S. Saminger-Platz, eds.), {Johannes} Kepler University, Linz, (2009), 70--73.

  \bibitem{li} S. G.  Li, X. Xin and Y. L. Li, {\it Closure axioms for a class of fuzzy matroids and co-towers of matroids},
 Fuzzy Sets and Systems, {\bf 158} (2007), 1246--1257.

 \bibitem{luo} X. L. Luo and J. M. Fang, {\it Fuzzifying closure systems and
  closure operators}, Iranian Journal of Fuzzy Systems, {\textbf 8}{\bf(1)} (2011), 77--94.

 \bibitem{mas} A. S. Mashhour and M. H. Ghanim, {\it Fuzzy closure spaces}, J. Math. Anal. Appl., {\bf 106} (1985), 154--170.

 \bibitem{rod1} S. E. Rodabaugh, {\it Powerset operator foundations for poslat fuzzy set theories and topologies}, in \cite{hoh}, 91--116.

\bibitem{rod2} S. E. Rodabaugh, {\it Categorical foundations of variable-basis fuzzy topology}, in \cite{hoh}, 273--388.

{\bibitem{rod3} S. E. Rodabaugh, {\it Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics}, Int. J. Math. Math. Sci., {\bf 2007(3)} (2007), 1--71.}

 \bibitem{shi} F. G. Shi, {\it $L$-fuzzy interiors and $L$-fuzzy
  closures}, Fuzzy Sets and Systems, {\bf 160} (2009), 1218--1232.

 \bibitem{Shi2} F. G. Shi, {\it Regularity and normality of $(L,M)$-fuzzy topological spaces}, Fuzzy Sets and Systems, {\bf 182} (2011), 37--52.

 \bibitem{sos} A. P. \v{S}ostak, {\it On a fuzzy topological structure},
  Rend. Circ. Mat. Palermo, (Supp. Ser. II), {\bf 11} (1985), 83--103 .

 \bibitem{sri1} R. Srivastava, A. K. Srivastava and A. Choubey, {\it Fuzzy closure
  spaces}, J. Fuzzy Math., {\bf 2} (1994), 525--534.

\bibitem{wang} G. J. Wang, {\it Theory of topological molecular
  lattices}, Fuzzy Sets and Systems, {\bf 47}{\bf(3)} (1992), 351--376.

  \bibitem{wangl} L. Wang and F. G. Shi,  {\it Charaterization of $L$-fuzzifying matroids by $L$-fuzzifying closure operators},
  Iranian Journal of Fuzzy Systems, {\bf 7} (2010), 47--58.

 \bibitem{yue} Y. L. Yue and J. M. Fang, {\it Categories isomorphic to the Kubiak-\u{S}ostak extension of
  TML}, Fuzzy Sets and Systems, {\bf 157} (2006), 832--842.

 \bibitem{zhou} W. Zhou, {\it  Generalization of $L$-closure spaces}, Fuzzy Sets and Systems, {\bf 149} (2005), 415--432.