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A new definition of order relation for the introduction of algebraic fuzzy closure operators. (English) Zbl 1423.03222

Summary: In this paper, a new approach to order relation between fuzzy sets is provided, which is called well inclusion order between fuzzy sets. Based on this new order relation, the concept of algebraic fuzzy closure operators is introduced. It is shown that there is a categorical isomorphism between algebraic fuzzy closure operators and fuzzy convex structures. Also, the relationship between fuzzy closure systems and fuzzy convex structures is investigated. It is proved that the category of fuzzy convex spaces is a bicoreflective subcategory of the category of fuzzy closure system spaces.

MSC:

03E72 Theory of fuzzy sets, etc.
06A15 Galois correspondences, closure operators (in relation to ordered sets)
03G25 Other algebras related to logic
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[1] Abramsky, S.; Jung, A., Domain theory, (Handbook of Logic in Computer Science, vol. 3, (1994)), 1-168
[2] Adámek, J.; Herrlich, H.; Strecker, G. E., Abstract and concrete categories, (1990), Wiley New York · Zbl 0695.18001
[3] Bandler, W.; Kohout, L., Special properties, closures and interiors of crisp and fuzzy relations, Fuzzy Sets Syst., 26, 3, 317-331, (1988) · Zbl 0664.04001
[4] Bělohlávek, R., Fuzzy relational systems: foundations and principles, (2002), Kluwer Academic Publishers, Plenum Publishers New York · Zbl 1067.03059
[5] Bělohlávek, R., Fuzzy closure operators, J. Math. Anal. Appl., 262, 473-489, (2001) · Zbl 0989.54006
[6] Bělohlávek, R.; De Baets, B.; Outrata, J.; Vychodil, V., Computing the lattice of all fixpoints of a fuzzy closure operator, IEEE Trans. Fuzzy Syst., 18, 3, 546-557, (2010)
[7] Biacino, L.; Gerla, G., Closure systems and L-subalgebras, Inf. Sci., 33, 181-195, (1984) · Zbl 0562.06004
[8] Biacino, L.; Gerla, G., An extension principle for closure operators, J. Math. Anal. Appl., 198, 1-24, (1996) · Zbl 0855.54007
[9] Bustince, H.; Galar, M.; Bedregal, B., A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications, IEEE Trans. Fuzzy Syst., 21, 6, 1150-1162, (2013)
[10] De Baets, B., Analytical solution methods for fuzzy relational equations, (Dubois, D.; Prade, H., Fundamentals of Fuzzy Sets, Handb. Fuzzy Sets Ser., vol. 1, (2000), Kluwer Norwell, MA), 291-340, ch. 6 · Zbl 0970.03044
[11] Fang, J. M., Stratified L-ordered convergence structures, Fuzzy Sets Syst., 161, 2130-2149, (2010) · Zbl 1197.54015
[12] Fang, J. M., Relationships between L-ordered convergence structures and strong L-topologies, Fuzzy Sets Syst., 161, 2923-2944, (2010) · Zbl 1271.54030
[13] Gerla, G., Graded consequence relations and fuzzy closure operators, J. Appl. Non-Class. Log., 6, 369-379, (1996) · Zbl 0872.03012
[14] Gerla, G., Fuzzy logic. mathematical tools for approximate reasoning, (2001), Kluwer Dordrecht, The Netherlands · Zbl 0976.03026
[15] Guo, L.; Zhang, G.; Li, Q., Fuzzy closure systems on L-ordered sets, Math. Log. Q., 57, 281-291, (2011) · Zbl 1218.06002
[16] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht, The Netherlands · Zbl 0937.03030
[17] Jang, L. C., Interval-valued Choquet integrals and their applications, J. Appl. Math. Comput., 16, 429-443, (2004)
[18] John, T.; Janet, A.; Greg, G., Fuzzy subsethood for fuzzy sets of type-2 and generalized type-n, IEEE Trans. Fuzzy Syst., 17, 1, 50-60, (2009)
[19] Klement, E. P.; Mesiar, R.; Pap, E., A universal integral as common frame for Choquet and sugeno integral, IEEE Trans. Fuzzy Syst., 18, 1, 178-187, (2010)
[20] Klir, G. J.; Yuan, B., Fuzzy sets and fuzzy logic. theory and applications, (1995), Prentice-Hall Englewood Cliffs, NJ · Zbl 0915.03001
[21] Li, L. Q.; Jin, Q., On adjunctions between lim, SL-top, and SL-lim, Fuzzy Sets Syst., 182, 66-78, (2011) · Zbl 1244.54018
[22] Maruyama, Y., Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 164, 22-37, (2009)
[23] Mendel, J.; John, R., Type-2 fuzzy sets made simple, IEEE Trans. Fuzzy Syst., 10, 2, 117-127, (2002)
[24] Murali, V., Lattice of fuzzy subalgebras and closure systems in \(I^X\), Fuzzy Sets Syst., 41, 101-111, (1991) · Zbl 0731.08007
[25] Pang, B.; Shi, F.-G., Subcategories of the category of L-convex spaces, Fuzzy Sets Syst., 313, 61-74, (2017) · Zbl 1372.52001
[26] Pang, B.; Zhao, Y., Characterizations of L-convex spaces, Iran. J. Fuzzy Syst., 13, 4, 51-61, (2016) · Zbl 1358.52002
[27] Preuss, G., Foundations of topology-an approach to convenient topology, (2002), Kluwer Academic Publisher Dordrecht, Boston, London · Zbl 1058.54001
[28] Rosa, M. V., On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets Syst., 62, 97-100, (1994) · Zbl 0854.54010
[29] Shi, F.-G.; Xiu, Z.-Y., A new approach to the fuzzification of convex structures, J. Appl. Math., 2014, (2014), 12 pp.
[30] Shi, F.-G.; Li, E.-Q., The restricted hull operator of M-fuzzifying convex structures, J. Intell. Fuzzy Syst., 30, 409-421, (2015) · Zbl 1364.54011
[31] Van De Vel, M., Theory of convex structures, (1993), North-Holland Amsterdam · Zbl 0785.52001
[32] Wu, X.-Y.; Bai, S.-Z., On M-fuzzifying JHC convex structures and M-fuzzifying Peano interval spaces, J. Intell. Fuzzy Syst., 30, 2447-2458, (2016) · Zbl 1364.52001
[33] Xiu, Z.-Y.; Shi, F.-G., M-fuzzifying interval spaces, Iran. J. Fuzzy Syst., 14, 1, 145-162, (2017) · Zbl 1370.54008
[34] Yao, W.; Zhao, B., Kernel systems on L-ordered sets, Fuzzy Sets Syst., 182, 101-109, (2011) · Zbl 1241.06004
[35] Zadeh, L. A., Fuzzy sets, Inf. Control, 8, 3, 338-353, (1965) · Zbl 0139.24606
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