Document Type : Research Paper


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

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


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


S. E. Abbas, {\it $(r,s)$-generalized intuitionistic fuzzy closed
sets},  J. Egypt Math. Soc., {\bf 14}\textbf{(2)} (2006), 283--297.
S. E. Abbas and Halis Ayg\"{u}n, {\it Intuitionistic fuzzy
semiregularization spaces}, Information Sciences, {\bf 176} (2006),
J. Adamek, H. Herrlich and G. E. Strecker, {\it Abstract and concrete
categories},  Wiley, New York, 1990.
K. Atanassov, {\it Intuitionistic fuzzy sets},  Fuzzy Sets and
Systems, \textbf{20}\textbf{(1)} (1986), 87-96.

 C. L.
Chang, {\it Fuzzy topological spaces}, J. Math. Anal. Appl.,
\textbf{24} (1968), 182-190.
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.
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.
D. \c{C}oker, {\it An introduction to intuitionistic fuzzy
topological spaces}, Fuzzy Sets and Systems, {\bf 88} (1997),
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.

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.

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

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

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.

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

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

U. H\"ohle and A. P. \v{S}ostak, {\it A general theory of fuzzy
topological spaces}, Fuzzy Sets and Systems, {\bf 73} (1995),

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).

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.

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.

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

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

T. Kubiak and A. P. \v{S}ostak, {\it Lower set-valued fuzzy
topologies}, Quaestiones Math., \textbf{20}\textbf{(3)} (1997),

 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.

Y. M. Liu and M. K. Luo, {\it Fuzzy topology}, Scientific Publishing
Co. Singapore, 1997.

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

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

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

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.

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.

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

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

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.

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

E. Turunen, {\it Mathematics behind fuzzy logic}, A Springer-Verlag
Co., New York, 1999.

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