\((L,M)\)-fuzzy convex structures
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Authors
Fu-Gui Shi
- School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China.
Zhen-Yu Xiu
- College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610000, China.
Abstract
In this paper, the notion of \((L,M)\)-fuzzy convex structures is introduced. It is a generalization of L-convex structures and
\(M\)-fuzzifying convex structures. In our definition of \((L,M)\)-fuzzy convex structures, each \(L\)-fuzzy subset can be regarded as an
\(L\)-convex set to some degree. The notion of convexity preserving functions is also generalized to lattice-valued case. Moreover,
under the framework of \((L,M)\)-fuzzy convex structures, the concepts of quotient structures, substructures and products are
presented and their fundamental properties are discussed. Finally, we create a functor \(\omega\) from MYCS to LMCS and show that
MYCS can be embedded in LMCS as a coreflective subcategory, where MYCS and LMCS denote the category of \(M\)-fuzzifying
convex structures and the category of \((L,M)\)-fuzzy convex structures, respectively.
Share and Cite
ISRP Style
Fu-Gui Shi, Zhen-Yu Xiu, \((L,M)\)-fuzzy convex structures, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3655--3669
AMA Style
Shi Fu-Gui, Xiu Zhen-Yu, \((L,M)\)-fuzzy convex structures. J. Nonlinear Sci. Appl. (2017); 10(7):3655--3669
Chicago/Turabian Style
Shi, Fu-Gui, Xiu, Zhen-Yu. "\((L,M)\)-fuzzy convex structures." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3655--3669
Keywords
- quotient structures
- substructures
- \((L،M)\)-fuzzy convex structure
- \((L،M)\)-fuzzy convexity preserving function
- products.
MSC
References
-
[1]
N. Ajmal, K. V. Thomas, Fuzzy lattices, Inform. Sci., 79 (1994), 271–291.
-
[2]
M. Berger, Convexity, Amer. Math. Monthly, 97 (1990), 650–678.
-
[3]
P. Dwinger, Characterization of the complete homomorphic images of a completely distributive complete lattice, I, Nederl. Akad. Wetensch. Indag. Math., 85 (1982), 403–414.
-
[4]
J.-M. Fang, P.-W. Chen, One-to-one correspondence between fuzzifying topologies and fuzzy preorders, Fuzzy Sets and Systems, 158 (2007), 1814–1822.
-
[5]
H.-L. Huang, F.-G. Shi, L-fuzzy numbers and their properties, Inform. Sci., 178 (2008), 1141–1151.
-
[6]
Q. Jin, L.-Q. Li , On the embedding of convex spaces in stratified L-convex spaces, SpringerPlus, 5 (2016), 10 pages.
-
[7]
F. Jinming, I-fuzzy Alexandrov topologies and specialization orders, Fuzzy Sets and Systems, 158 (2007), 2359–2374.
-
[8]
T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz University, Poznan, Poland (1985)
-
[9]
M. Lassak, On metric B-convexity for which diameters of any set and its hull are equal, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25 (1977), 969–975.
-
[10]
Y. Maruyama, Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 1641 (2009), 22–37.
-
[11]
C. V. Negoiţă, D. A. Ralescu, Applications of fuzzy sets to systems analysis, Translated from the Romanian, ISR— Interdisciplinary Systems Research, Birkhäuser Verlag, Basel-Stuttgart, 11 (1975), 187 pages.
-
[12]
B. Pang, F.-G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets and Systems, 313 (2017), 61–74.
-
[13]
B. Pang, Y. Zhao, Characterizations of L-convex spaces, Iran. J. Fuzzy Syst., 13 (2016), 51–61.
-
[14]
M. V. Rosa, A study of fuzzy convexity with special reference to separation properties, Ph.D. Thesis, Cochin University of Science and Technology, Kerala, India (1994)
-
[15]
M. V. Rosa, On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets and Systems, 62 (1994), 97–100.
-
[16]
F.-G. Shi, Theory of \(L_\beta\)-nested sets and \(L_\alpha\)-nested sets and its applications, (Chinese) Fuzzy Syst. Math., 4 (1995), 65–72.
-
[17]
F.-G. Shi, L-fuzzy relations and L-fuzzy subgroups, J. Fuzzy Math., 8 (2000), 491–499.
-
[18]
F.-G. Shi, E.-Q. Li, The restricted hull operator of M-fuzzifying convex structures, J. Intell. Fuzzy Syst., 30 (2015), 409–421.
-
[19]
F.-G. Shi, Z.-Y. Xiu, A new approach to the fuzzification of convex structures, J. Appl. Math., 2014 (2014), 12 pages.
-
[20]
V. P. Soltan, d-convexity in graphs, (Russian) Dokl. Akad. Nauk SSSR, 272 (1983), 535–537.
-
[21]
A. P. Šostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo, 11 (1985), 89–103.
-
[22]
M. L. J. van de Vel, Theory of convex structures, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam (1993)
-
[23]
J. van Mill, Supercompactness and Wallman spaces, Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam (1977)
-
[24]
J. C. Varlet, Remarks on distributive lattices, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 23 (1975), 1143–1147.
-
[25]
G.-J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47 (1992), 351–376.
-
[26]
Z.-Y. Xiu, F.-G. Shi, M-fuzzifying interval spaces, Iran. J. Fuzzy Syst., 14 (2017), 145–162.
-
[27]
M.-S. Ying, A new approach for fuzzy topology, I, Fuzzy Sets and Systems, 39 (1991), 303–321.
-
[28]
L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353.