A survey of categorical properties of \(\mathbb{L}\)-fuzzy relations. (English) Zbl 1522.18004
Summary: An \(\mathbb{L}\)-fuzzy relation is a relation valued on a complete lattice \(\mathbb{L}\) with a monoidal structure. This paper reviews four categories of \(\mathbb{L}\)-fuzzy relations each modelling an area where Fuzzy Set Theory can be applied. We review the notions of these multivalued binary relations and present some basic properties of the corresponding categories aiming at applications in areas such as Computing Science, Linear Logic and Quantum Mechanics. The emphasis is on the monoidal aspects of the categories. Monoidal categories are one of the most applied kinds of categories. A monoidal view of Fuzzy Relations may widen the spectrum of applications of Fuzzy Set Theory.
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| [1] | Goguen, J. A., L-fuzzy sets, J. Math. Anal. Appl., 18, 1, 145-174 (1967) · Zbl 0145.24404 |
| [2] | Street, R., Monoidal categories in, and linking, geometry and algebra, Bull. Belg. Math. Soc. Simon Stevin, 19, 5, 769-820 (2012) · Zbl 1267.18006 |
| [3] | Seely, R. A., Linear Logic,*-Autonomous Categories and Cofree Coalgebras (1987), CEGEP John Abbott College: CEGEP John Abbott College Ste. Anne de Bellevue, Quebec · Zbl 0674.03007 |
| [4] | A. Schalk, What is a categorical model of linear logic, Manuscript, available from http://www.cs.man.ac.uk/schalk/work.html. |
| [5] | Blute, R.; Scott, P., Category theory for linear logicians, (Linear Logic in Computer Science, vol. 316 (2004)), 3-64 · Zbl 1068.03057 |
| [6] | Hotz, G., Eine algebraisierung des syntheseproblems von schaltkreisen I, Elektron. Inf.verarb. Kybern., 1, 3, 185-205 (1965) · Zbl 0156.25504 |
| [7] | Fong, B., The algebra of open and interconnected systems, arXiv preprint |
| [8] | Baez, J. C.; Fong, B., A compositional framework for passive linear networks, arXiv preprint · Zbl 1402.18005 |
| [9] | Fong, B.; Spivak, D.; Tuyéras, R., Backprop as functor: a compositional perspective on supervised learning, (2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science. 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS (2019), IEEE), 1-13 |
| [10] | Ghani, N.; Hedges, J.; Winschel, V.; Zahn, P., Compositional game theory, (Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (2018)), 472-481 · Zbl 1452.91016 |
| [11] | de Felice, G.; Toumi, A.; Coecke, B., Discopy: monoidal categories in Python, arXiv preprint |
| [12] | Bonchi, F.; Sobociński, P.; Zanasi, F., A categorical semantics of signal flow graphs, (International Conference on Concurrency Theory (2014), Springer), 435-450 · Zbl 1417.68119 |
| [13] | Riley, M., Categories of optics, arXiv preprint |
| [14] | Brown, C.; Hutton, G., Categories, allegories and circuit design, (Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science (1994), IEEE), 372-381 |
| [15] | Abramsky, S., Retracing some paths in process algebra, (International Conference on Concurrency Theory (1996), Springer), 1-17 · Zbl 1514.68153 |
| [16] | Abramsky, S.; Coecke, B., A categorical semantics of quantum protocols, (Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004 (2004), IEEE), 415-425 |
| [17] | Coecke, B.; Paquette, E. O.; Pavlovic, D., Classical and quantum structuralism, (Semantic Techniques in Quantum Computation (2008)), 29-69 · Zbl 1192.81016 |
| [18] | Pavlovic, D., Relating toy models of quantum computation: comprehension, complementarity and dagger mix autonomous categories, Electron. Notes Theor. Comput. Sci., 270, 2, 121-139 (2011) · Zbl 1348.81174 |
| [19] | Zimmermann, H.-J., Fuzzy Set Theory—and Its Applications (2011), Springer Science & Business Media |
| [20] | Kruse, R.; Gebhardt, J. E.; Klowon, F., Foundations of Fuzzy Systems (1994), John Wiley & Sons, Inc. · Zbl 0843.68109 |
| [21] | Yen, J.; Langari, R., Fuzzy Logic: Intelligence, Control, and Information, vol. 1 (1999), Prentice Hall: Prentice Hall Upper Saddle River, NJ |
| [22] | Pykacz, J., Fuzzy sets in foundations of quantum mechanics, (On Fuzziness (2013), Springer), 553-557 |
| [23] | del Castillo-Mussot, M.; Dias, R. C., Fuzzy sets and physics, Rev. Mex. Fis., 39, 2, 295-303 (1992) · Zbl 1291.03108 |
| [24] | Schmitt, I.; Nürnberger, A.; Lehrack, S., On the relation between fuzzy and quantum logic, (Views on Fuzzy Sets and Systems from Different Perspectives (2009), Springer), 417-438 · Zbl 1170.03332 |
| [25] | De, S. K.; Biswas, R.; Roy, A. R., An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets Syst., 117, 2, 209-213 (2001) · Zbl 0980.92013 |
| [26] | Gen, M.; Tsujimura, Y.; Zheng, D., An application of fuzzy set theory to inventory control models, Comput. Ind. Eng., 33, 3-4, 553-556 (1997) |
| [27] | Lee, H.-M., Applying fuzzy set theory to evaluate the rate of aggregative risk in software development, Fuzzy Sets Syst., 79, 3, 323-336 (1996) |
| [28] | Barr, M., Fuzzy set theory and topos theory, Can. Math. Bull., 29, 4, 501-508 (1986) · Zbl 0563.03040 |
| [29] | Walker, C. L., Categories of fuzzy sets, Soft Comput., 8, 4, 299-304 (2004) · Zbl 1071.18004 |
| [30] | Korniłowicz, A., Cartesian products of relations and relational structures, Formaliz. Math., 6, 1, 145-152 (1997) |
| [31] | Corradini, A.; Asperti, A., A categorical model for logic programs: indexed monoidal categories, (Workshop/School/Symposium of the REX Project (Research and Education in Concurrent Systems) (1992), Springer), 110-137 |
| [32] | Haghverdi, E.; Scott, P., A categorical model for the geometry of interaction, Theor. Comput. Sci., 350, 2, 252-274 (2006) · Zbl 1086.03050 |
| [33] | Coecke, B.; Edwards, B., Toy quantum categories, Electron. Notes Theor. Comput. Sci., 270, 1, 29-40 (2011) · Zbl 1348.81049 |
| [34] | Barr, M., Fuzzy models of linear logic, Math. Struct. Comput. Sci., 6, 3, 301-312 (1996) · Zbl 0856.03049 |
| [35] | Schalk, A.; De Paiva, V., Poset-valued sets or how to build models for linear logics, Theor. Comput. Sci., 315, 1, 83-107 (2004) · Zbl 1055.03040 |
| [36] | Kawahara, Y.; Furusawa, H., An algebraic formalization of fuzzy relations, Fuzzy Sets Syst., 101, 1, 125-135 (1999) · Zbl 0934.03068 |
| [37] | Winter, M., Goguen categories, J. Relat. Methods Comput. Sci., 1, 339-357 (2004) |
| [38] | Winter, M., Goguen Categories: A Categorical Approach to L-Fuzzy Relations, vol. 25 (2007), Springer Science & Business Media · Zbl 1261.03154 |
| [39] | Coecke, B.; Pavlovic, D., Quantum measurements without sums, (Mathematics of Quantum Computing and Technology (2007)) |
| [40] | Marsden, D.; Genovese, F., Custom hypergraph categories via generalized relations, arXiv preprint · Zbl 1433.68205 |
| [41] | Breuvart, F., The resource lambda calculus is short-sighted in its relational model, (International Conference on Typed Lambda Calculi and Applications (2013), Springer), 93-108 · Zbl 1381.03021 |
| [42] | Harding, J.; Walker, C.; Walker, E., Categories with fuzzy sets and relations, Fuzzy Sets Syst., 256, 149-165 (2014) · Zbl 1335.03051 |
| [43] | Coecke, B., New Structures for Physics, vol. 813 (2011), Springer · Zbl 1200.00038 |
| [44] | Winter, M.; Jackson, E., Categories of relations for variable-basis fuzziness, Fuzzy Sets Syst., 298, 222-237 (2016) · Zbl 1386.18004 |
| [45] | Hovey, M., Model Categories, Mathematical Surveys and Monographs, vol. 63 (1999) · Zbl 0909.55001 |
| [46] | Hussain, M., Fuzzy relations (2010), Blekinge Institute of Technology School of Engineering, Department of Mathematics and Science, Master’s thesis |
| [47] | Lee, K. H., First Course on Fuzzy Theory and Applications, vol. 27 (2006), Springer Science & Business Media |
| [48] | Awodey, S., Category Theory (2010), Oxford University Press · Zbl 1194.18001 |
| [49] | Abramsky, S.; Heunen, C., H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics, Clifford Lect., 71, 1-24 (2012) · Zbl 1267.18007 |
| [50] | Kleisli, H., Every standard construction is induced by a pair of adjoint functors, Proc. Am. Math. Soc., 16, 3, 544-546 (1965) · Zbl 0138.01704 |
| [51] | Fiore, M.; Menni, M., Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads, Theory Appl. Categ., 15, 2, 40-65 (2005) · Zbl 1073.18001 |
| [52] | Szigeti, J., On limits and colimits in the Kleisli category, Cah. Topol. Géom. Différ. Catég., 24, 4, 381-391 (1983) · Zbl 0533.18006 |
| [53] | Power, J.; Robinson, E., Premonoidal categories and notions of computation, Math. Struct. Comput. Sci., 7, 5, 453-468 (1997) · Zbl 0897.18002 |
| [54] | Kock, A., Monads on symmetric monoidal closed categories, Arch. Math., 21, 1, 1-10 (1970) · Zbl 0196.03403 |
| [55] | Heunen, C.; Karvonen, M., Monads on dagger categories, arXiv preprint · Zbl 1378.18003 |
| [56] | Heunen, C.; Karvonen, M., Reversible monadic computing, Electron. Notes Theor. Comput. Sci., 319, 217-237 (2015) · Zbl 1351.68100 |
| [57] | Karvonen, M., The way of the dagger, preprint |
| [58] | Jenčová, A.; Jenča, G., On monoids in the category of sets and relations, Int. J. Theor. Phys., 56, 12, 3757-3769 (2017) · Zbl 1387.81022 |
| [59] | Arbib, M. A.; Manes, E. G., Fuzzy machines in a category, Bull. Aust. Math. Soc., 13, 2, 169-210 (1975) · Zbl 0318.18008 |
| [60] | Power, J., Models for the computational λ-calculus, Electron. Notes Theor. Comput. Sci., 40, 288-301 (2001) · Zbl 1264.03047 |
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