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Categorically-algebraic topology and its applications | ||
| Iranian Journal of Fuzzy Systems | ||
| مقاله 6، دوره 12، شماره 3، شهریور 2015، صفحه 57-94 اصل مقاله (610.05 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22111/ijfs.2015.2020 | ||
| نویسنده | ||
| Sergey A. Solovyov* | ||
| Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2896/2, 616 69 Brno, Czech Republic and Institute of Mathematics and Computer Science, University of Latvia, Raina bulvaris 29, LV-1459 Riga, Latvia | ||
| چکیده | ||
| This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respective categories of topological structures are topological over their ground categories. The theory also extends the notion of topological system of S.~Vickers (and its numerous many-valued modifications of J.~T.~Denniston, A.~Melton and S.~E.~Rodabaugh), and shows that the categories of catalg topological structures are isomorphic to coreflective subcategories of the categories of catalg topological systems. This extension initiates a new approach to soft topology, induced by the concept of soft set of D.~Molodtsov, and currently pursued by various researchers. | ||
| کلیدواژهها | ||
| Categorically-algebraic topology؛ lattice-valued topology؛ Soft topology؛ Topological category؛ Topological system؛ Topological theory | ||
| مراجع | ||
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[1] J. Adamek, H. Herrlich, and G. E. Strecker, Abstract and concrete categories: the joy of cats, Dover Publications (Mineola, New York), 2009. [2] J. Adamek, J. Rosicky, and E. M. Vitale, Algebraic theories. a categorical introduction to general algebra, Cambridge University Press, 2011. [3] D. Aerts, Foundations of quantum physics: a general realistic and operational approach, Int. J. Theor. Phys., 38(1) (1999), 289-358. [4] D. Aerts, E. Colebunders, A. van der Voorde and B. van Steirteghem, State property systems and closure spaces: a study of categorical equivalence, Int. J. Theor. Phys., 38(1) (1999), 359-385. [5] D. Aerts, E. Colebunders, A. van der Voorde and B. van Steirteghem, On the amnestic modication of the category of state property systems, Appl. Categ. Struct., 10(5) (2002), 469-480. [6] H. Aktas and N. C agman, Soft sets and soft groups, Inf. Sci., 177(13) (2007), 2726-2735. [7] J. M. Anthony and H. Sherwood, Fuzzy groups redened, J. Math. Anal. Appl., 69(1) (1979), 124-130. [8] B. Banaschewski and E. Nelson, Tensor products and bimorphisms, Canad. Math. Bull., 19(4) (1976), 385-402. [9] M. Barr and C. Wells, Toposes, triples and theories, Repr. Theory Appl. Categ., 2005(12) (2005), 1-288. [10] G. Birkho, On the structure of abstract algebras, Proc. Cambridge Phil. Soc., 31(4) (1935), 433-454. [11] S. Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, Springer-Verlag, 78 (1981). [12] M. Caldas, S. Jafari and R. K. Saraf, Semi--open sets and new classes of maps, Bull. Iran. Math. Soc., 31(2) (2005), 37-52. [13] N. C agman, S. Karatas and S. Enginoglu, Soft topology, Comput. Math. Appl., 62(1) (2011), 351-358. [14] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24(1) (1968), 182-190. [15] M. M. Clementino, D. Hofmann and W. Tholen, One setting for all: Metric, topology, uniformity, approach structure, Appl. Categ. Struct., 12(2) (2004), 127-154. [16] P. M. Cohn, Universal Algebra, D. Reidel Publ. Comp., 1981. [17] C. De Mitri and C. Guido, Some remarks on fuzzy powerset operators, Fuzzy Sets Syst., 126(2) (2002), 241-251. [18] M. Demirci, Pointed semi-quantales and lattice-valued topological spaces, Fuzzy Sets Syst., 161(9) (2010), 1224-1241. [19] J. T. Denniston, A. Melton and S. E. Rodabaugh, Lattice-valued topological systems, Abstracts of the 30th Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universitat, Linz, (2009), 24-31. [20] J. T. Denniston, A. Melton and S. E. Rodabaugh, Lattice-valued predicate transformers and interchange systems, Abstracts of the 31st Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universitat, Linz, (2010), 31-40. [21] J. T. Denniston, A. Melton and S. E. Rodabaugh, Formal concept analysis and lattice-valued interchange systems, Abstracts of the 32nd Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universitat, Linz, (2011), 41-47. [22] J. T. Denniston, A. Melton and S. E. Rodabaugh, Interweaving algebra and topology: Lattice-valued topological systems, Fuzzy Sets Syst., 192 (2012), 58-103. [23] J. T. Denniston and S. E. Rodabaugh, Functorial relationships between lattice-valued topol- ogy and topological systems, Quaest. Math., 32(2) (2009), 139-186. [24] A. Di Nola and G. Gerla, Lattice valued algebras, Stochastica, 11(2-3) (1987), 137-150. [25] Y. Diers, Categories of algebraic sets, Appl. Categ. Struct., 4(2-3) (1996), 329-341. [26] Y. Diers, Ane algebraic sets relative to an algebraic theory, J. Geom., 65(1-2) (1999), 54-76. [27] Y. Diers, Topological geometrical categories, J. Pure Appl. Algebra, 168(2-3) (2002), 177- 187. [28] D. Dikranjan, E. Giuli and A. Tozzi, Topological categories and closure operators, Quaest. Math., 11(3) (1988), 323-337. [29] C. Ehresmann, Gattungen von lokalen Strukturen, Jahresber. Dtsch. Math.-Ver., (German), 60(1) (1957), 49-77. [30] P. Eklund, Categorical fuzzy topology, Ph.D. thesis, Abo Akademi, 1986. [31] P. Eklund, M. A. Galan and W. Gahler, Partially ordered monads for monadic topologies, rough sets and Kleene algebras, Electron. Notes Theor. Comput. Sci., 225 (2009), 67-81. [32] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl., 56(10) (2008), 2621-2628. [33] A. Frascella, Attachment and topological systems in varieties of algebras, Ph.D. thesis, Department of Mathematics \Ennio De Giorgi", University of Salento, Italy, 2011. [34] A. Frascella, C. Guido and S. Solovyov, Dual attachment pairs in categorically-algebraic topology, Appl. Gen. Topol., 12(2) (2011), 101-134. [35] W. Gahler, Monadic topology { a new concept of generalized topology, Recent Developments of General Topology and its Applications, International Conference in Memory of Felix Hausdor (1868 - 1942) (W. Gahler, ed.), Akademie-Verlag, (1992), 136-149. [36] W. Gahler, The general fuzzy lter approach to fuzzy topology I, Fuzzy Sets Syst., 76(2) (1995), 205-224. [37] W. Gahler, The general fuzzy lter approach to fuzzy topology II, Fuzzy Sets Syst., 76(2) (1995), 225-246. [38] W. Gahler, General topology { the monadic case, examples, applications, Acta Math. Hung., 88(4) (2000), 279-290. [39] B. Ganter and R. Wille, Formale begrisanalyse. mathematische grundlagen, Berlin: Springer, 1996. [40] G. Gierz, K. H. Hofmann and etc., Continuous lattices and domains, Cambridge University Press, 2003. [41] J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18(1) (1967), 145-174. [42] J. A. Goguen, The fuzzy Tychono theorem, J. Math. Anal. Appl., 43(3) (1973), 734-742. [43] G. Gratzer, Universal Algebra, 2nd ed., Springer, 2008. [44] C. Guido, The subspace problem in the traditional point-set context of fuzzy topology, Quaest. Math., 20(3) (1997), 351-372. [45] C. Guido, Powerset operators based approach to fuzzy topologies on fuzzy sets, Topological and Algebraic Structures in Fuzzy Sets. A Handbook of Recent Developments in the Mathematics of Fuzzy Sets (S. E. Rodabaugh and E. P. Klement, eds.), Kluwer Academic Publishers, (2003), 401-413. [46] C. Guido, Fuzzy points and attachment, Fuzzy Sets Syst., 161(16) (2010), 2150-2165. [47] C. Guido and V. Scarciglia, L-topological spaces as spaces of points, Fuzzy Sets Syst., 173(1) (2011), 45-59. [48] C. Guido and S. Solovyov, Topological systems versus attachment relations, Quaest. Math., 37(4) (2014), 455-484. [49] H. Herrlich and G. E. Strecker, Category theory, 3rd ed., Sigma Series in Pure Mathematics, Heldermann Verlag, 1 (2007). [50] U. Hohle, Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78(2) (1980), 659-673. [51] U. Hohle, A note on the hypergraph functor, Fuzzy Sets Syst., 131(3) (2002), 353-356. [52] U. Hohle and A. P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (U. Hohle and S. E. Rodabaugh, eds.), Kluwer Academic Publishers, (1999), 123-272. [53] S. Hussain and B. Ahmad, Some properties of soft topological spaces, Comput. Math. Appl., 62(11) (2011), 4058-4067. [54] B. Hutton, Products of fuzzy topological spaces, Topology Appl., 11(1) (1980), 59-67. [55] J. R. Isbell, Atomless parts of spaces, Math. Scand., 31 (1972), 5-32. [56] P. T. Johnstone, Stone spaces, Cambridge University Press, 1982. [57] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl., 56(5) (2008), 1408-1413. [58] J. C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc., 13(III) (1963), 71-89. [59] W. Kotze and T. Kubiak, Fuzzy topologies of Scott continuous functions and their relation to the hypergraph functor, Quaest. Math., 15(2) (1992), 175-187. [60] T. Kubiak, On fuzzy topologies, Ph.D. thesis, Adam Mickiewicz University, Poznan, Poland, 1985. [61] T. Kubiak and A. Sostak, Foundations of the theory of (L;M)-fuzzy topological spaces, Abstracts of the 30th Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universitat, Linz, (2009), 70-73. [62] F. W. Lawvere, Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories, Repr. Theory Appl. Categ., 2004(5) (2004), 1-121. [63] N. Levine, Semi-open sets and semi-continuity in topological spaces, Am. Math. Mon., 70 (1963), 36-41. [64] F. E. J. Linton, Some aspects of equational categories, Proc. Conf. Categor. Algebra, La Jolla, (1965), 84{94. [65] X. Liu, D. Xiang, J. Zhan and K. P. Shum, Isomorphism theorems for soft rings, Algebra Colloq., 19(4) (2012), 649-656. [66] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56(3) (1976), 621-633. [67] S. Mac Lane, Categories for the working mathematician, 2nd ed., Springer-Verlag, 1998. [68] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl., 45(4-5) (2003), 555-562. [69] E. G. Manes, Algebraic theories, Springer-Verlag, 1976. [70] W. K. Min, A note on soft topological spaces, Comput. Math. Appl., 62(9) (2011), 3524- 3528. [71] D. Molodtsov, Soft set theory { rst results, Comput. Math. Appl., 37(4-5) (1999), 19-31. [72] J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, Singapore: World Scientic, 1998. [73] C. J. Mulvey and J. W. Pelletier, On the quantisation of points, J. Pure Appl. Algebra, 159(2) (2001), 231-295. [74] C. J. Mulvey and J. W. Pelletier, On the quantisation of spaces, J. Pure Appl. Algebra, 175(1-3) (2002), 289-325. [75] N. Nakajima, Generalized fuzzy sets, Fuzzy Sets Syst., 32(3) (1989), 307-314. [76] C. V. Negoita and D. A. Ralescu, Applications of fuzzy sets to systems analysis, Interdisciplinary Systems Research, Birkhauser Verlag, 11 (1975). [77] D. Papert and S. Papert, Sur les treillis des ouverts et les paratopologies, Semin. de Topologie et de Geometrie dierentielle Ch. Ehresmann 1 (1957/58), 1 (1959), 1-9. [78] B. Pazar Varol, A. Shostak and H. Aygun, Categories related to topology viewed as soft sets, Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011) and LFA-2011, Atlantis Press, 1 (2011), 883-890. [79] V. Pratt, Chu spaces, School on Category Theory and Applications (Coimbra, 1999), 39-100, Textos Mat. Ser. B, 21, Univ. Coimbra, Coimbra, 1999. [80] G. Richter, Kategorielle Algebra, Akademie-Verlag, 1979. [81] S. E. Rodabaugh, The Hausdor separation axiom for fuzzy topological spaces, Topology Appl., 11(3) (1980), 319-334. [82] S. E. Rodabaugh, A categorical accommodation of various notions of fuzzy topology, Fuzzy Sets Syst., 9(1) (1983), 241-265. [83] S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets Syst., 40(2) (1991), 297- 345. [84] S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (U. Hohle and S. E. Rodabaugh, eds.), Kluwer Academic Publishers, (1999), 273-388. [85] S. E. Rodabaugh, Necessary and sucient conditions for powersets in Set and Set C to form algebraic theories, Abstracts of the 26th Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universitat, Linz, (2005), 89-97. [86] S. E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topolog- ical theories for lattice-valued mathematics, Int. J. Math. Math. Sci., 2007 (2007), 1-71. [87] S. E. Rodabaugh, Functorial comparisons of bitopology with topology and the case for redun- dancy of bitopology in lattice-valued mathematics, Appl. Gen. Topol., 9(1) (2008), 77-108. [88] S. E. Rodabaugh, Relationship of algebraic theories to powersets over objects in Set and SetC, Fuzzy Sets Syst., 161(3) (2010), 453-470. [89] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35(3) (1971), 512-517. [90] K. I. Rosenthal, Quantales and their applications, Addison Wesley Longman, 1990. [91] J. Rosicky, Equational categories, Cah. Topol. Geom. Dier., 22(1) (1981), 85{95. [92] G. J. Seal, A Kleisli-based approach to lax algebras, Appl. Categ. Structures, 17(1) (2009), 75-89. [93] M. Shabir and M. Naz, On soft topological spaces, Comput. Math. Appl., 61(7) (2011), 1786-1799. [94] F.-G. Shi and B. Pang, Redundancy of fuzzy soft topological spaces, J. Intell. Fuzzy Syst., 27(4) (2014), 1757-1760. [95] S. Solovyov, On the category Set(JCPos), Fuzzy Sets Syst., 157(3) (2006), 459-465. [96] S. Solovyov, Categorical frameworks for variable-basis sobriety and spatiality, Math. Stud. (Tartu), 4 (2008), 89-103. [97] S. Solovyov, Sobriety and spatiality in varieties of algebras, Fuzzy Sets Syst., 159(19) (2008), 2567-2585. [98] S. Solovyov, Variable-basis topological systems versus variable-basis topological spaces, Soft Comput., 14(10) (2010), 1059-1068. [99] S. Solovyov, Fuzzy algebras as a framework for fuzzy topology, Fuzzy Sets Syst., 173(1) (2011), 81-99. [100] S. Solovyov, On a generalization of the concept of state property system, Soft Comput., 15(12) (2011), 2467-2478. [101] S. Solovyov, Categorical foundations of variety-based topology and topological systems, Fuzzy Sets Syst., 192 (2012), 176-200. [102] S. Solovyov, Composite variety-based topological theories, Fuzzy Sets Syst., 195 (2012), 1-32. [103] S. Solovyov, Categorically-algebraic topology versus universal topology, Fuzzy Sets Syst., 227 (2013), 25-45. [104] S. Solovyov, Lattice-valued soft algebras, Soft Comput., 17(10) (2013), 1751-1766. [105] A. P. Sostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo, II. Ser. Suppl., 11 (1985), 89-103. [106] S. Vickers, Topology via Logic, Cambridge University Press, 1989. [107] L. A. Zadeh, Fuzzy sets, Inf. Control, 8(3) (1965), 338-365. [108] D. Zhang and Y.-M. Liu, L-fuzzy version of Stone's representation theorem for distributive lattices, Fuzzy Sets Syst., 76(2) (1995), 259-270. | ||
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