×

\(I\)-fuzzy Alexandrov topologies and specialization orders. (English) Zbl 1136.54002

The aim of this paper is to investigate the relationships between \(I\)-fuzzy structures and fuzzy order structures on a universe set \(X\); in this paper an \(I\)-fuzzy topology is a map \(\tau: I^X\to [0,1]\) satisfying the following conditions
1) \(\tau(\lambda)= 1\) for all constant functions \(\lambda\) from \(X\) to \([0, 1]\);
2) \(\tau(u\wedge v)\geq \tau(u)\wedge \tau(v)\), \(\forall u,v\in I^X\);
3) \(\tau(\bigvee_{j\in J} u_j)\geq \bigwedge_{j\in J}\tau(u_j)\), \(\forall j\in J\), \(u_j\in I^X\);
If a map \(\tau: 2^X\to [0,1]\) satisfies similar conditions as an \(I\)-fuzzy topology on a set \(X\), then \(T\) is called a fuzzyfying topology.
With \(X\) a universe set and ‘\(*\)’ a \(t\)-norm on \([0, 1]\), a fuzzy relation \(R: X\times X\to[0,1]\) is said to be a \(*\)-fuzzy preorder if it is a reflexive and \(*\)-transitive fuzzy relation.
The author focuses on the problem how to obtain an \(I\)-fuzzy topology through a fuzzy relation on a set. The notion of an \(I\)-fuzzified set of all upper sets (dually, an \(I\)-fuzzyfied set of all lower sets) of a fuzzy preordered set \((X, R)\) is introduced and it is shown that an \(I\)-fuzzyfied set of all upper sets induced by a fuzzy preorder \(R\) is precisely an \(I\)-fuzzy topology of the fuzzy preordered set \((X,R)\); this topology is named an \(I\)-fuzzy Alexandrov topology in this paper.
Further, for a given \(I\)-fuzzy topological space \((X,\tau)\), the author considers the problem of inducing a fuzzy order on \(X\) by \(\tau\); in this context the author proposes a definition of the specialization order of an \(I\)-fuzzy topological space.
Finally, the author gives a ‘Representation theorem of fuzzy preorders by \(I\)-fuzzy topologies’.

MSC:

54A40 Fuzzy topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adamek, J.; Herrlich, H.; Strecker, G. E., Abstract and Concrete Categories (1990), Wiley: Wiley New York · Zbl 0695.18001
[2] Bodenhofer, U.; De Baets, B.; Fodor, J., A compendium of fuzzy weak orders: representations and characterizations, Fuzzy Sets and Systems, 158, 811-829 (2007) · Zbl 1119.06001
[3] Bodenhofer, U.; De Cock, M.; Kerre, E. E., Openings and closures of fuzzy orderings: theoretical basis and applications to fuzzy rule-based systems, Internat. J. Gen. Systems, 32, 343-360 (2003) · Zbl 1110.03048
[4] J.M. Fang, P.W. Chen, One-to-one correspondence between fuzzifying topologies and fuzzy preorders, Fuzzy Sets and Systems, accepted for publication.; J.M. Fang, P.W. Chen, One-to-one correspondence between fuzzifying topologies and fuzzy preorders, Fuzzy Sets and Systems, accepted for publication. · Zbl 1137.54005
[5] Höhle, U., Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78, 659-673 (1980) · Zbl 0462.54002
[6] U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999.; U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999.
[7] Höhle, U.; Šostak, A., Axiomatic foundations of variable-basis fuzzy topology, (Höhle, U.; Rodabaugh, S. E., Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3 (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, Dordrecht, London), 123-272 · Zbl 0977.54006
[8] Jacas, J.; Recasens, J., Fixed points and generators of fuzzy relations, J. Math. Anal. Appl., 186, 21-29 (1994) · Zbl 0807.04003
[9] Jacas, J.; Recasens, J., Fuzzy \(T\)-transitive relations: eigenvectors and generators, Fuzzy Sets and Systems, 72, 147-154 (1995) · Zbl 0844.04006
[10] Kelley, J. L., General Topology, Graduate Text in Mathematics, Vol. 27 (1975), Springer: Springer Berlin
[11] Klawonn, F.; Castro, J. L., Similarity in fuzzy reasoning, Math. Soft. Comput., 2, 197-228 (1995) · Zbl 0859.04006
[12] Klement, E. P.; Mesiar, R.; Pap, E., Triangular norms, (Trends in Logic, Vol. 8 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht) · Zbl 1007.20054
[13] J. Kortelainen, Some compositional modifier operators generate \(L\)-interior operators, in: Proceedings of 10th IFSA World Congress, 2003, pp. 480-483.; J. Kortelainen, Some compositional modifier operators generate \(L\)-interior operators, in: Proceedings of 10th IFSA World Congress, 2003, pp. 480-483.
[14] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.; T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985. · Zbl 0589.54013
[15] Lai, H.; Zhang, D., Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157, 1865-1885 (2006) · Zbl 1118.54008
[16] Lowen, R., Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56, 623-633 (1976) · Zbl 0342.54003
[17] Lowen, R., Initial and final fuzzy topologies and the fuzzy Tychonoff theorem, J. Math. Anal. Appl., 58, 11-21 (1977) · Zbl 0347.54002
[18] Qin, K.; Pei, Z., On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems, 151, 605-613 (2005) · Zbl 1070.54006
[19] Rodabaugh, S. E., Categorical foundations of variable-basis fuzzy topology, (Höhle, U.; Rodabaugh, S. E., Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3 (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, Dordrecht, London), 273-388 · Zbl 0968.54003
[20] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland Amsterdam · Zbl 0546.60010
[21] Šostak, A., Two decades of fuzzy topology: basic ideas notions and results, Russian Math. Surveys, 44, 125-186 (1989) · Zbl 0716.54004
[22] Valverde, L., On the structure of F-indistinguishability operators, Fuzzy Sets and Systems, 17, 313-328 (1985) · Zbl 0609.04002
[23] Ying, M. S., A new approach to fuzzy topology (I), Fuzzy Sets and Systems, 39, 3, 303-321 (1991) · Zbl 0718.54017
[24] Zhang, D. X., \(L\)-Fuzzifying topologies as \(L\)-topologies, Fuzzy Sets and Systems, 125, 135-144 (2002) · Zbl 0992.54008
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.