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Fuzzy Galois connections on fuzzy sets. (English) Zbl 1397.03075

Summary: In fairly elementary terms, this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by several key results from the theory of quantaloid-enriched categories, this paper develops all needed ingredients purely in order-theoretic languages for the readership of fuzzy set theorists, with particular attention paid to fuzzy Galois connections between preordered fuzzy sets.

MSC:

03E72 Theory of fuzzy sets, etc.
06A75 Generalizations of ordered sets
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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[1] Adámek, J.; Herrlich, H.; Strecker, G. E., Abstract and concrete categories: the joy of cats, (1990), Wiley New York · Zbl 0695.18001
[2] Bělohlávek, R., Fuzzy Galois connections, Math. Log. Q., 45, 4, 497-504, (1999) · Zbl 0938.03079
[3] Bělohlávek, R., Fuzzy relational systems: foundations and principles, IFSR International Series on Systems Science and Engineering, vol. 20, (2002), Kluwer Academic Publishers Dordrecht · Zbl 1067.03059
[4] Bělohlávek, R., Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic, 128, 1-3, 277-298, (2004) · Zbl 1060.03040
[5] Betti, R.; Carboni, A., Cauchy-completion and the associated sheaf, Cah. Topol. Géom. Différ. Catég., 23, 3, 243-256, (1982) · Zbl 0496.18008
[6] Betti, R.; Carboni, A.; Street, R.; Walters, R. F.C., Variation through enrichment, J. Pure Appl. Algebra, 29, 2, 109-127, (1983) · Zbl 0571.18004
[7] Birkhoff, G., Lattice theory, American Mathematical Society Colloquium Publications, vol. 25, (1967), American Mathematical Society Providence · Zbl 0126.03801
[8] Borceux, F.; Cruciani, R., Skew ω-sets coincide with ω-posets, Cah. Topol. Géom. Différ. Catég., 39, 3, 205-220, (1998) · Zbl 0917.18001
[9] Bukatin, M.; Kopperman, R.; Matthews, S. G.; Pajoohesh, H., Partial metric spaces, Am. Math. Mon., 116, 8, 708-718, (2009) · Zbl 1229.54037
[10] Denniston, J. T.; Melton, A.; Rodabaugh, S. E., Enriched categories and many-valued preorders: categorical, semantical, and topological perspectives, Fuzzy Sets Syst., 256, 4-56, (2014) · Zbl 1335.68305
[11] Erné, M.; Koslowski, J.; Melton, A.; Strecker, G. E., A primer on Galois connections, Ann. N.Y. Acad. Sci., 704, 1, 103-125, (1993) · Zbl 0809.06006
[12] Fourman, M. P.; Scott, D. S., Sheaves and logic, (Fourman, M. P.; Mulvey, C. J.; Scott, D. S., Applications of Sheaves: Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Durham, July 9-21, 1977, Lecture Notes in Mathematics, vol. 753, (1979), Springer Berlin, Heidelberg), 302-401
[13] Georgescu, G.; Popescu, A., Non-commutative fuzzy Galois connections, Soft Comput., 7, 7, 458-467, (2003) · Zbl 1024.03025
[14] Georgescu, G.; Popescu, A., Non-dual fuzzy connections, Arch. Math. Log., 43, 1009-1039, (2004) · Zbl 1060.03042
[15] Gutiérrez García, J.; Mardones-Pérez, I.; de Prada Vicente, M. A.; Zhang, D., Fuzzy Galois connections categorically, Math. Log. Q., 56, 2, 131-147, (2010) · Zbl 1206.06002
[16] Herrlich, H.; Hušek, M., Galois connections categorically, J. Pure Appl. Algebra, 68, 1, 165-180, (1990) · Zbl 0718.18001
[17] Heymans, H., Sheaves on quantales as generalized metric spaces, (2010), Universiteit Antwerpen Belgium, PhD thesis
[18] (Hofmann, D.; Seal, G. J.; Tholen, W., Monoidal Topology: A Categorical Approach to Order, Metric, and Topology, Encyclopedia of Mathematics and Its Applications, vol. 153, (2014), Cambridge University Press Cambridge) · Zbl 1297.18001
[19] Höhle, U., Many-valued preorders I: the basis of many-valued mathematics, (Magdalena, L.; Verdegay, J. L.; Esteva, F., Enric Trillas: A Passion for Fuzzy Sets: A Collection of Recent Works on Fuzzy Logic, Studies in Fuzziness and Soft Computing, vol. 322, (2015), Springer Cham), 125-150 · Zbl 1383.06001
[20] Höhle, U., Modules in the category sup, (Saminger-Platz, S.; Mesiar, R., On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory, Studies in Fuzziness and Soft Computing, vol. 336, (2016), Springer Cham), 23-56
[21] Höhle, U.; Blanchard, N., Partial ordering in L-underdeterminate sets, Inf. Sci., 35, 2, 133-144, (1985) · Zbl 0576.06004
[22] Höhle, U.; Kubiak, T., A non-commutative and non-idempotent theory of quantale sets, Fuzzy Sets Syst., 166, 1-43, (2011) · Zbl 1226.06011
[23] Kelly, G. M., Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, vol. 64, (1982), Cambridge University Press Cambridge · Zbl 0478.18005
[24] Künzi, H.-P. A.; Pajoohesh, H.; Schellekens, M. P., Partial quasi-metrics, Theor. Comput. Sci., 365, 3, 237-246, (2006) · Zbl 1109.54021
[25] Lai, H.; Shen, L., Fixed points of adjoint functors enriched in a quantaloid, Fuzzy Sets Syst., 321, 1-28, (2017) · Zbl 1390.18006
[26] Lai, H.; Zhang, D., Fuzzy preorder and fuzzy topology, Fuzzy Sets Syst., 157, 14, 1865-1885, (2006) · Zbl 1118.54008
[27] Lai, H.; Zhang, D., Complete and directed complete ω-categories, Theor. Comput. Sci., 388, 1-25, (2007) · Zbl 1131.18006
[28] Lai, H.; Zhang, D., Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory, Int. J. Approx. Reason., 50, 5, 695-707, (2009) · Zbl 1191.68658
[29] Lawvere, F. W., Metric spaces, generalized logic and closed categories, Rend. Semin. Mat. Fis. Milano, XLIII, 135-166, (1973) · Zbl 0335.18006
[30] Li, W.; Lai, H.; Zhang, D., Yoneda completeness and flat completeness of ordered fuzzy sets, Fuzzy Sets Syst., 313, 1-24, (2017) · Zbl 1393.03035
[31] Mac Lane, S., Categories for the working Mathematician, Graduate Texts in Mathematics, vol. 5, (1998), Springer New York · Zbl 0906.18001
[32] Matthews, S. G., Partial metric topology, Ann. N.Y. Acad. Sci., 728, 1, 183-197, (1994) · Zbl 0911.54025
[33] Ore, O., Galois connexions, Trans. Am. Math. Soc., 55, 3, 493-513, (1944) · Zbl 0060.06204
[34] Pu, Q.; Zhang, D., Preordered sets valued in a GL-monoid, Fuzzy Sets Syst., 187, 1, 1-32, (2012) · Zbl 1262.18008
[35] Rosenthal, K. I., Quantales and their applications, Pitman Research Notes in Mathematics Series, vol. 234, (1990), Longman Harlow · Zbl 0703.06007
[36] Rosenthal, K. I., The theory of quantaloids, Pitman Research Notes in Mathematics Series, vol. 348, (1996), Longman Harlow · Zbl 0845.18003
[37] Shen, L., Adjunctions in quantaloid-enriched categories, (2014), Sichuan University Chengdu, PhD thesis
[38] Shen, L., \(\mathcal{Q}\)-closure spaces, Fuzzy Sets Syst., 300, 102-133, (2016) · Zbl 1378.54015
[39] Shen, L.; Tholen, W., Limits and colimits of quantaloid-enriched categories and their distributors, Cah. Topol. Géom. Différ. Catég., 56, 3, 209-231, (2015) · Zbl 1334.18005
[40] Shen, L.; Tholen, W., Topological categories, quantaloids and isbell adjunctions, Topol. Appl., 200, 212-236, (2016) · Zbl 1333.18009
[41] Shen, L.; Zhang, D., The concept lattice functors, Int. J. Approx. Reason., 54, 1, 166-183, (2013) · Zbl 1288.06013
[42] Shen, L.; Zhang, D., Categories enriched over a quantaloid: isbell adjunctions and kan adjunctions, Theory Appl. Categ., 28, 20, 577-615, (2013) · Zbl 1273.18022
[43] Shen, L.; Zhang, D., Formal concept analysis on fuzzy sets, (Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), (2013)), 215-219
[44] Stubbe, I., Categorical structures enriched in a quantaloid: categories, distributors and functors, Theory Appl. Categ., 14, 1, 1-45, (2005) · Zbl 1079.18005
[45] Stubbe, I., Categorical structures enriched in a quantaloid: tensored and cotensored categories, Theory Appl. Categ., 16, 14, 283-306, (2006) · Zbl 1119.18005
[46] Stubbe, I., An introduction to quantaloid-enriched categories, Fuzzy Sets Syst., 256, 95-116, (2014) · Zbl 1335.18002
[47] Stubbe, I., The double power monad is the composite power monad, Fuzzy Sets Syst., 313, 25-42, (2017) · Zbl 1390.18020
[48] Tao, Y.; Lai, H.; Zhang, D., Quantale-valued preorders: globalization and cocompleteness, Fuzzy Sets Syst., 256, 236-251, (2014) · Zbl 1337.06010
[49] Wagner, K. R., Solving recursive domain equations with enriched categories, (1994), Carnegie Mellon University Pittsburgh, PhD thesis
[50] Walters, R. F.C., Sheaves and Cauchy-complete categories, Cah. Topol. Géom. Différ. Catég., 22, 3, 283-286, (1981) · Zbl 0495.18009
[51] Yao, W.; Lu, L.-X., Fuzzy Galois connections on fuzzy posets, Math. Log. Q., 55, 1, 105-112, (2009) · Zbl 1172.06001
[52] Zadeh, L. A., Similarity relations and fuzzy orderings, Inf. Sci., 3, 2, 177-200, (1971) · Zbl 0218.02058
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