ON (L;M)-FUZZY CLOSURE SPACES

Document Type: Research Paper

Authors

1 Department of Mathematics, Kocaeli University, 41380, Kocaeli, Turkey.

2 Department of Mathematics, Kocaeli University, 41380, Kocaeli, Turkey.

3 Department of Mathematics, Faculty of Science, Sohag 82524, Egypt.

Abstract

The aim of this paper is to introduce $(L,M)$-fuzzy closure
structure where $L$ and $M$ are strictly two-sided, commutative
quantales. Firstly, we define $(L,M)$-fuzzy closure spaces and get
some relations between $(L,M)$-double fuzzy topological spaces and
$(L,M)$-fuzzy closure spaces. Then, we introduce initial
$(L,M)$-fuzzy closure structures and we prove that the category
$(L,M)$-{bf FC} of $(L,M)$-fuzzy closure spaces and
$(L,M)$-$mathcal{C}$-maps is a topological category over the
category {bf SET}. From this fact, we define products of
$(L,M)$-fuzzy closure spaces. Finally, we show that an initial
structure of $(L,M)$-double fuzzy topological spaces can be obtained
by the initial structure of $(L,M)$-fuzzy closure spaces induced by
them.

Keywords


bibitem{Sab:Gifcs}
S. E. Abbas, {it $(r,s)$-generalized intuitionistic fuzzy closed
sets},  J. Egypt Math. Soc., {bf 14}textbf{(2)} (2006), 283--297.

bibitem{SabHa:Ifss}
S. E. Abbas and Halis Ayg"{u}n, {it Intuitionistic fuzzy
semiregularization spaces}, Information Sciences, {bf 176} (2006),
745--757.

bibitem{Ahs:Acc}
J. Adamek, H. Herrlich and G. E. Strecker, {it Abstract and concrete
categories},  Wiley, New York, 1990.

bibitem{Kat:Ifs}
K. Atanassov, {it Intuitionistic fuzzy sets},  Fuzzy Sets and
Systems, textbf{20}textbf{(1)} (1986), 87-96.


bibitem{Clc:Fts}
 C. L.
Chang, {it Fuzzy topological spaces}, J. Math. Anal. Appl.,
textbf{24} (1968), 182-190.

bibitem{Chs:Goft}
K. C. Chattopadhyay, R. N. Hazra and S. K. Samanta, {it Gradation of
openness: fuzzy topology}, Fuzzy Sets and Systems, textbf{49}
(1992), 237-242.

bibitem{CS:Ftfco}
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{Dc:Iifts}
D. c{C}oker, {it An introduction to intuitionistic fuzzy
topological spaces}, Fuzzy Sets and Systems, {bf 88} (1997),
81-89.

bibitem{DcMd:Ifts}
D. c{C}oker and M. Demirci, {it An introduction to intuitionistic
fuzzy topological spaces in v{S}ostak sense}, Busefal, {bf 67}
(1996), 67-76.


bibitem{JgSr:Ott}
J. Gutierrez Garcia and S. E. Rodabaugh, {it Order-theoretic
topological, categorical redundancies of interval-valued sets, grey
sets, vague sets, interval-valued intuitionistic sets,
intuitionistic fuzzy sets and topologies}, Fuzzy Sets and Systems,
{bf 156} (2005), 445-484.


bibitem{Han:Scifts}
I. M. Hanafy, A. M. Abd El-Aziz and T. M. Salman, {it Semi
I-compactness in intuitionistic fuzzy topological spaces}, Iranian
Journal of Fuzzy Systems, textbf{3}textbf{(2)} (2006), 53-62.

bibitem{Hoh:Usfs}
U. H"ohle, {it Upper semicontinuous fuzzy sets and applications},
J. Math. Anall. Appl., {bf 78} (1980), 659-673.


bibitem{Hoh:Mcct}
U. H"ohle, {it  Monoidal closed categories, weak topoi and
generalized logics}, Fuzzy Sets and Systems, {bf 42} (1991), 15-35.

bibitem{Hoh:Mss}
U. H"ohle, {it  M-valued sets and sheaves over integral
commutative cl-monoids, in Applications of category theory of fuzzy
subsets (S. Rodabaugh, E. P. Klement and U. H"ohle, eds.)}, Kluwer
Academic, Dordrecht, Boston, (1992), 33-72.


bibitem{Hoh:Crm}
U. H"ohle, {it Commutative, residuated l-monoids}, Non-classical
logics and their Applications to Fuzzy Subsets theory (Linz, 1992),
Kluwer, Acad. Publ., Dordrecht, (1995), 53-106.


bibitem{Hoh:Mvt}
U. H"ohle, {it Many valued topology and its applications}, Kluwer
Academic Publisher, Boston, 2001.


bibitem{HohKl:Ncl}
U. H"ohle and E. P. Klement, {it Non-classical logic and their
applications to fuzzy subsets}, Kluwer Academic Publisher, Boston, 1995.


bibitem{HohSos:Gtfts}
U. H"ohle and A. P. v{S}ostak, {it A general theory of fuzzy
topological spaces}, Fuzzy Sets and Systems, {bf 73} (1995),
131-149.


bibitem{HohSos:Affbft}
U. H"ohle and A. P. v{S}ostak, {it Axiomatic foundations of
fixed-basis fuzzy topology}, The Handbooks of Fuzzy Sets Series,
 Kluwer Academic Publishers, Dordrecht (Chapter 3), {bf3} (1999).


bibitem{Sj:Sgm}
S. Jenei, {it Structure of Girard monoids on [0,1]}, Chapter 10,
In: S. E. Rodabaugh, E. P. Klement, eds., Topological and Algebraic
Structures in Fuzzy Sets, Kluwer Academic Publ., 2003.


bibitem{YcKYsK:As}
Y. C. Kim and Y. S. Kim, {it $(L,odot)$-approximation spaces and
$(L,odot)$-fuzzy quasi-uniform spaces}, Information Sciences, {bf 179}
(2009), 2028-2048.


bibitem{YcKYmK:Ipf}
Y. C. Kim and J. M. Ko, {it Images and preimages of L-filterbases},
Fuzzy Sets and Systems, {bf 157} (2006), 1913-1927.


bibitem{Tk:Ft}
T. Kubiak, {it On fuzzy topologies}, Ph. D. Thesis, A. Mickiewicz,
Poznan, 1985.


bibitem{TkSos:Lsft}
T. Kubiak and A. P. v{S}ostak, {it Lower set-valued fuzzy
topologies}, Quaestiones Math., textbf{20}textbf{(3)} (1997),
423-429.


bibitem{EplY:Mfts}
 E. P. Lee and Y. B. Im, {it Mated
fuzzy topological spaces}, J. Korea Fuzzy Logic Intell. Sys. Soc.,
textbf{11}textbf{(2)} (2001), 161-165.


bibitem{Ym:Ft}
Y. M. Liu and M. K. Luo, {it Fuzzy topology}, Scientific Publishing
Co. Singapore, 1997.


bibitem{Rl:Ftsfc}
 R. Lowen, {it Fuzzy
topological spaces and fuzzy compactness}, J. Math. Anal. Appl.,
textbf {56} (1976), 621-633.


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


bibitem{Mul:Q}
 C. J.
Mulvey, {it $&$ }, Suppl. Rend. Circ.Mat. Palermo Ser. II,
textbf{12} (1986), 99-104.


bibitem{SeR:Cfv}
S. E. Rodabaugh, {it Categorical foundations of variable-basis
topology, in U. Hohle, S. E. Rodabaugh, eds., mathematics of fuzzy
sets: logic, topology and measure theory, the handbooks of fuzzy
sets series}, Kluwer Academic publishers, Dordrecht, textbf{3}
(1999), 273-388.


bibitem{SeREpk:Tas}
S. E. Rodabaugh and E. P. Klement, {it Topological and algebraic
structures in fuzzy sets}, The Handbook of Recent Developments in
the Mathematics of Fuzzy Sets, Trends in Logic 20, Kluwer Academic
Publishers, (Boston/Dordrecht/London), 2003.


bibitem{SksTkm:Igo}
S. K. Samanta and T. K. Mondal, {it On intuitionistic gradation of
openness}, Fuzzy Sets and Systems, textbf{131} (2002), 323-336.


bibitem{FgS:Cclp}
F. G. Shi, {it Countable compactness and the lindelof property of
L-fuzzy sets}, Iranian Journal of Fuzzy Systems, textbf{1}textbf{(1)} (2004), 79-88.

bibitem{ASos:Ofts}
A. P. v{S}ostak, {it On a fuzzy topological structure}, Suppl.
Rend. Circ. Matem. Palerms ser II, textbf{11} (1985), 89-103.


bibitem{Asos:Dft}
A. P. v{S}ostak, {it Two decades of fuzzy topology: basic ideas,
notions and results}, Russian Math. Surveys, textbf{44}
textbf{(6)} (1989), 125-186.


bibitem{Asos:Bsft}
A. P. v{S}ostak, {it Basic structures of fuzzy topology}, J. Math.
Sci., textbf{78}textbf{(6)} (1996), 662-701.


bibitem{Et:Mbfl}
E. Turunen, {it Mathematics behind fuzzy logic}, A Springer-Verlag
Co., New York, 1999.


bibitem{Msy:Nafft}
M. S. Ying, {it A new approach for fuzzy topology (I)}, Fuzzy Sets
and Systems, textbf{39} (1991), 303-320.