×

Fuzzy preorder and fuzzy topology. (English) Zbl 1118.54008

The paper deals mainly with fuzzy preorder; to be more specific, the categorical aspects of the interrelationship between fuzzy preorder, topological spaces, and fuzzy topological spaces is investigated.
The authors delineate basic properties of continuous t-norms and concrete adjoint functors at the beginning; with a brief review on the connection between topological spaces and preordered sets in section 2, section 3 gives a systematic investigation of the properties of upper sets and preordered sets along with several examples. Finally, a fuzzy topology \(\Gamma^*(R)\) is constructed on \(X\) for every fuzzy preordered set \((X,R)\), where \(\Gamma^*(R)\) is the Alexandrov topology generated by \(R\). On the other hand, for every fuzzy topological space \((X,\tau)\), a fuzzy preorder \(\Omega^*(\tau)\) on \(X\) is constructed; \(\Omega^*(\tau)\) is the specialization order on \((X,\tau)\). It is shown that these two constructions are functorial and compatible with their classical counterparts and that the functors \(\Gamma^*\) and \(\Omega^*\) form a pair of adjoint functors between the category of fuzzy preordered sets and that of fuzzy topological spaces.

MSC:

54A40 Fuzzy topology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Adámek, J.; Herrlich, H.; Strecker, G.E., Abstract and concrete categories, (1990), Wiley New York · Zbl 0695.18001
[2] Bodenhofer, U., Representations and constructions of similarity-based fuzzy orderings, Fuzzy sets and systems, 137, 113-136, (2003) · Zbl 1052.91032
[3] Bodenhofer, U.; De Cock, M.; Kerre, E.E., Openings and closures of fuzzy preorderings: theoretical basis and applications to fuzzy rule-based systems, Internat. J. general systems, 32, 343-360, (2003) · Zbl 1110.03048
[4] Boixader, D.; Jacas, J.; Recasens, J., Upper and lower approximations of fuzzy sets, J. general systems, 29, 555-568, (2000) · Zbl 0955.03056
[5] F. Borceux, Handbook of Categorical Algebra, vol. 2, Categories and Structures, Cambridge University Press, Cambridge, 1994. · Zbl 0843.18001
[6] Castro, J.L.; Delgado, M.; Trillas, E., Inducing implication relations, Internat. J. approx. reasoning, 10, 235-250, (1994) · Zbl 0809.68095
[7] De Baets, B.; Mesiar, R., Metrics and \(\mathcal{T}\)-equalities, J. math. anal. appl., 267, 531-547, (2002) · Zbl 0996.03035
[8] De Cock, M.; Kerre, E.E., On (un)suitable fuzzy relations to model approximate equality, Fuzzy sets and systems, 133, 137-153, (2003) · Zbl 1020.03049
[9] Demirci, M., Topological properties of the class of generators of an indistinguishability operator, Fuzzy sets and systems, 143, 413-426, (2004) · Zbl 1057.54007
[10] Elorza, J.; Burillo, P., Connecting fuzzy preorders, fuzzy consequence operators and fuzzy closure and co-closure systems, Fuzzy sets and systems, 139, 601-613, (2003) · Zbl 1032.03021
[11] Herrlich, H.; Hušek, M., Galois connections categorically, J. pure appl. algebra, 68, 165-180, (1990) · Zbl 0718.18001
[12] U. Höhle, Many-valued equalities and their representations, Preprint, 2004.
[13] Höhle, U.; Šostak, A.P., Axiomatic foundations of fixed-basis fuzzy topology, (), 123-272 · Zbl 0977.54006
[14] Jacas, J.; Recasens, J., Fixed points and generators of fuzzy relations, J. math. anal. appl., 186, 21-29, (1994) · Zbl 0807.04003
[15] Jacas, J.; Recasens, J., Fuzzy T-transitive relations: eigenvectors and generators, Fuzzy sets and systems, 82, 147-154, (1995) · Zbl 0844.04006
[16] Jenei, S., New family of triangular norms via contrapositive symmetrization of residuated implications, Fuzzy sets and systems, 110, 157-174, (2000) · Zbl 0941.03059
[17] Johnstone, P.T., Stone spaces, (1982), Cambridge University Press Cambridge · Zbl 0499.54001
[18] Kelley, J.L., General topology, graduate text in mathematics, vol. 27, (1975), Springer Berlin
[19] Klawonn, F.; Castro, J.L., Similarity in fuzzy reasoning, Mathware soft comput., 2, 197-228, (1995) · Zbl 0859.04006
[20] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, trends in logic, vol. 8, (2000), Kluwer Academic Publishers Dordrecht
[21] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms. position paper II: general constructions and parametrized families, Fuzzy sets and systems, 145, 411-438, (2004) · Zbl 1059.03012
[22] Kortelainen, J., Some compositional modifier operators generate L-interior operators, (), 480-483
[23] Lawvere, F.W., Metric spaces, generalized logic, and closed categories, Rendi. semi. maté. fisico milano, 43, 135-166, (1973) · Zbl 0335.18006
[24] Lowen, E.; Lowen, R.; Wuyts, P., The categorical topology approach to fuzzy topology and fuzzy convergence, Fuzzy sets and systems, 40, 347-373, (1991) · Zbl 0728.54001
[25] Lowen, R., Fuzzy topological spaces and fuzzy compactness, J. math. anal. appl., 56, 623-633, (1976) · Zbl 0342.54003
[26] Mac Lane, S., Categories for the working Mathematician, (1998), Springer Berlin · Zbl 0906.18001
[27] Novák, V.; Perfilieva, I.; Močkoř, J., Mathematical principles of fuzzy logic, (1999), Kluwer Academic Publishers Boston/Dordrecht/London · Zbl 0940.03028
[28] Qin, K.; Pei, Z., On the topological properties of fuzzy rough sets, Fuzzy sets and systems, 151, 601-613, (2005) · Zbl 1070.54006
[29] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), North-Holland Amsterdam · Zbl 0546.60010
[30] Srivastava, A.K.; Tiwari, S.P., On relationships among fuzzy approximation operators fuzzy, topology, and fuzzy automata, Fuzzy sets and systems, 138, 197-204, (2003) · Zbl 1043.54004
[31] Valverde, L., On the structure of \(F\)-indistinguishability operators, Fuzzy sets and systems, 17, 313-328, (1985) · Zbl 0609.04002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.