تعداد نشریات | 27 |
تعداد شمارهها | 604 |
تعداد مقالات | 6,158 |
تعداد مشاهده مقاله | 9,090,347 |
تعداد دریافت فایل اصل مقاله | 5,933,018 |
ON ALGEBRAIC AND COALGEBRAIC CATEGORIES OF VARIETY-BASED TOPOLOGICAL SYSTEMS | ||
Iranian Journal of Fuzzy Systems | ||
مقاله 3، دوره 8، شماره 5، دی 2011، صفحه 13-30 اصل مقاله (432.89 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22111/ijfs.2011.295 | ||
نویسنده | ||
Sergey A. Solovyov ![]() ![]() | ||
Department of Mathematics, University of Latvia, Zellu iela 8, LV-1002 Riga, Latvia and Institute of Mathematics and Computer Science, University of Latvia, Raina bulvaris 29, LV-1459 Riga, Latvia | ||
چکیده | ||
Motivated by the recent study on categorical properties of latticevalued topology, the paper considers a generalization of the notion of topological system introduced by S. Vickers, providing an algebraic and a coalgebraic category of the new structures. As a result, the nature of the category TopSys of S. Vickers gets clari ed, and a metatheorem is stated, claiming that (latticevalued) topology can be embedded into algebra. | ||
کلیدواژهها | ||
Algebra؛ (Co)algebraic category؛ (Co)re ective subcategory؛ Latticevalued topology؛ Powerset operator؛ S-quantale؛ Topological category؛ Topological system؛ Variety | ||
مراجع | ||
\bibitem{Adamek2009} J.~Ad{\'{a}}mek, H.~Herrlich and G.~E. Strecker, {\it Abstract and Concrete Categories: the joy of cats}, Dover Publications, New York, 2009. \bibitem{Aerts1999} D.~Aerts, {\it Foundations of quantum physics: a general realistic and operational approach}, Int. J. Theor. Phys., {\bf 38}\textbf{(1)} (1999), 289-358. \bibitem{Aerts1999a} D.~Aerts, E.~Colebunders, A.~van~der Voorde and B.~van Steirteghem, {\it State property systems and closure spaces: a study of categorical equivalence}, Int. J. Theor. Phys., {\bf 38}\textbf{(1)} (1999), 359-385. \bibitem{Aerts2002} D.~Aerts, E.~Colebunders, A.~van~der Voorde and B.~van Steirteghem, {\it On the amnestic modification of the category of state property systems}, Appl. Categ. Struct., {\bf 10}\textbf{(5)} (2002), 469-480. \bibitem{Birkhoff1935} G.~Birkhoff, {\it On the structure of abstract algebras}, Proc. Cambridge Phil. Soc., {\bf 31} (1935), 433-454. \bibitem{Chang1968} C.~L. Chang, {\it Fuzzy topological spaces}, J. Math. Anal. Appl., {\bf 24} (1968), 182-190. \bibitem{Cohn1981} P.~M. Cohn, {\it Universal algebra}, D. Reidel Publ. Comp., 1981. \bibitem{Demirci2010} M.~Demirci, {\it Pointed semi-quantales and lattice-valued topological spaces}, Fuzzy Sets and Systems, {\bf 161}\textbf{(9)} (2010), 1224-1241. \bibitem{Denniston2009} J.~T. Denniston, A.~Melton and S.~E. Rodabaugh, {\it Lattice-valued topological systems}, In: U.~Bodenhofer, B.~De~Baets, E.~P. Klement, S.~Saminger-Platz, eds., Abstracts of the 30th Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universit\"{a}t, Linz, (2009), 24-31. \bibitem{Denniston2009a} J.~T. Denniston and S.~E. Rodabaugh, {\it Functorial relationships between lattice-valued topology and topological systems}, Quaest. Math., {\bf 32}\textbf{(2)} (2009), 139-186. \bibitem{Dikranjan1988} D.~Dikranjan, E.~Giuli and A.~Toi, {\it Topological categories and closure operators}, Quaest. Math., {\bf 11}\textbf{(3)} (1988), 323-337. \bibitem{Eklund1986} P.~Eklund, {\it Categorical fuzzy topology}, Ph.D. Thesis, \r{A}bo Akademi, 1986. \bibitem{Gahler1992} W.~G\"ahler, {\it Monadic topology-a new concept of generalized topology}, In: W.~G\"ahler, eds., Recent Developments of General Topology and Its Applications, Akademie-Verlag, Berlin, (1992), 136-149. \bibitem{Goguen1973} J.~A.~Goguen, {\it The fuzzy Tychonoff theorem}, J. Math. Anal. Appl., {\bf 43} (1973), 734-742. \bibitem{Guido2010} C.~Guido, {\it Fuzzy points and attachment}, Fuzzy Sets and Systems, {\bf 161}\textbf{(16)} (2010), 2150-2165. \bibitem{Guido2003} C.~Guido, {\it Powerset operators based approach to fuzzy topologies on fuzzy sets}, In: S.~E. Rodabaugh, E.~P. Klement, eds., Topological and Algebraic Structures in Fuzzy Sets, Kluwer Academic Publishers, (2003), 401-413. \bibitem{Guido2011} C.~Guido and V.~Scarciglia, {\it $L$-topological spaces as spaces of points}, Fuzzy Sets and Systems, {\bf173(1)} (2011), 45-59. \bibitem{Hohle1999a} U.~H\"{o}hle and A.~P. \v{S}ostak, {\it Axiomatic foundations of fixed-basis fuzzy topology}, In: U.~H\"{o}hle, S.~E. Rodabaugh, eds., Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, (1999), 123-272. \bibitem{Hutton1980} B.~Hutton, {\it Products of fuzzy topological spaces}, Topology Appl., {\bf 11} (1980), 59-67. \bibitem{Isbell1972} J.~R. Isbell, {\it Atomless parts of spaces}, Math. Scand., {\bf 31} (1972), 5-32. \bibitem{John2008} J.~John and T.~Baiju, {\it Metacompactness in $L$-topological spaces}, Iranian Journal of Fuzzy Systems, {\bf 5}\textbf{(3)} (2008), 71-79. \bibitem{Johnstone1982} P.~T. Johnstone, {\it Stone spaces}, Cambridge University Press, 1982. \bibitem{Kelley1975} J.~L. Kelley, {\it General topology}, Springer, 1975. \bibitem{Kruml2008} D.~Kruml and J.~Paseka, {\it Algebraic and categorical aspects of quantales}, In: M.~Hazewinkel, eds., Handbook of Algebra, Elsevier, {\bf5} (2008), 323-362. \bibitem{Li2005} S. P. Li, Z.~Fang and J.~Zhao, \emph{{P2-connectedness in $L$-topological spaces}}, Iranian Journal of Fuzzy Systems, {\bf 2}\textbf{(1)} (2005), 29-36. \bibitem{Lowen1976} R.~Lowen, {\it Fuzzy topological spaces and fuzzy compactness}, J. Math. Anal. Appl., {\bf 56} (1976), 621-633. \bibitem{MacLane1998} S.~{Mac Lane}, {\it Categories for the working mathematician}, 2nd ed., Springer-Verlag, 1998. \bibitem{Manes1976} E.~G. Manes, {\it Algebraic theories}, Springer-Verlag, 1976. \bibitem{Papert1959} D.~Papert and S.~Papert, {\it Sur les treillis des ouverts et les paratopologies}, Semin. de Topologie et de Geometrie differentielle Ch. Ehresmann 1 (1957/58), {\bf 1} (1959), 1-9. \bibitem{Pratt1999} V.~Pratt, {\it Chu spaces}, Coimbra: Universidade de Coimbra, Departamento de Matem\'atica, Textos Mat., S\'er., {\bf B}\textbf{(21)} (1999), 39-100. \bibitem{Pultr2003b} A.~Pultr, {\it Frames}, In: M.~Hazewinkel, eds., Handbook of Algebra, North-Holland Elsevier, {\bf3} (2003), 789-858. \bibitem{Pultr2003} A.~Pultr and S.~E. Rodabaugh, {\it Lattice-valued frames, functor categories, and classes of sober spaces}, In: S.~E. Rodabaugh, E.~P. Klement, eds., Topological and Algebraic Structures in Fuzzy Sets, Kluwer Academic Publishers, (2003), 153-187. \bibitem{Pultr2008} A.~Pultr and S.~E. Rodabaugh, {\it Category theoretic aspects of chain-valued frames: part I: categorical foundations}, Fuzzy Sets and Systems, {\bf 159}\textbf{(5)} (2008), 501-528. \bibitem{Pultr2008a} A.~Pultr and S.~E. Rodabaugh, {\it Category theoretic aspects of chain-valued frames: part II: applications to lattice-valued topology}, Fuzzy Sets and Systems, {\bf 159}\textbf{(5)} (2008), 529-558. \bibitem{Rodabaugh1983} S.~E. Rodabaugh, {\it A categorical accommodation of various notions of fuzzy topology}, Fuzzy Sets and Systems, {\bf 9} (1983), 241-265. \bibitem{Rodabaugh1997} S.~E. Rodabaugh, {\it Powerset operator based foundation for point-set lattice-theoretic (poslat) fuzzy set theories and topologies}, Quaest. Math., {\bf 20}\textbf{(3)} (1997), 463-530. \bibitem{Rodabaugh1999a} S.~E. Rodabaugh, {\it Categorical foundations of variable-basis fuzzy topology}, In: U.~H\"{o}hle, S.~E. Rodabaugh, eds., Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, (1999), 273-388. \bibitem{Rodabaugh1999c} S.~E. Rodabaugh, {\it Powerset operator foundations for poslat fuzzy set theories and topologies}, In: U.~H\"{o}hle, S.~E. Rodabaugh, eds., Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, (1999), 91-116. \bibitem{Rodabaugh2007} S.~E. Rodabaugh, {\it Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics}, Int. J. Math. Math. Sci., {\bf 2007} (2007), 1-71. \bibitem{Rosenthal1990} K.~I. Rosenthal, {\it Quantales and their applications}, Addison Wesley Longman, 1990. \bibitem{Shi2004} F. G. Shi, {\it Countable compactness and the Lindel\"{o}f property of $L$-fuzzy sets}, Iranian Journal of Fuzzy Systems, {\bf 1}\textbf{(1)} (2004), 79-88. \bibitem{Shi2010} F. G. Shi and P. Chen, {\it The Urysohn axiom and the completely Hausdorff axiom in $L$-topological spaces}, Iranian Journal of Fuzzy Systems, {\bf 7}\textbf{(1)} (2010), 33-45. \bibitem{Shostak1989} A.~P.~Shostak, {\it Two decades of fuzzy topology: basic ideas, notions, and results}, Russ. Math. Surv., {\bf 44}\textbf{(6)} (1989), 125-186. \bibitem{Solovjovs2008a} S.~Solovjovs, {\it On a categorical generalization of the concept of fuzzy set: basic definitions, properties, examples}, VDM Verlag Dr. M\"{u}ller, 2008. \bibitem{Solovjovs2009} S.~Solovjovs, {\it Embedding topology into algebra}, In: U.~Bodenhofer, B.~De~Baets, E.~P. Klement, S.~Saminger-Platz, eds., Abstracts of the 30th Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universit\"{a}t, Linz, (2009), 106-110. \bibitem{Solovyov2010c} S.~Solovyov, {\it Localification of variable-basis topological systems}, Quaest. Math., {\bf34(1)} (2011), 11-33. \bibitem{Solovyovl} S.~Solovyov, {\it On a generalization of the concept of state property system}, Soft Comput., to appear. \bibitem{Solovyovq} S.~Solovyov, {\it Powerset operator foundations for catalg fuzzy set theories}, Iranian Journal of Fuzzy Systems, {\bf 8(2)} (2011), 1-46. \bibitem{Solovyov2008d} S.~Solovyov, {\it Categorical frameworks for variable-basis sobriety and spatiality}, Math. Stud. Tartu, {\bf 4} (2008), 89-103. \bibitem{Solovyov2008b} S.~Solovyov, {\it Sobriety and spatiality in varieties of algebras}, Fuzzy Sets and Systems, {\bf 159}\textbf{(19)} (2008), 2567-2585. \bibitem{Solovyov2010a} S.~Solovyov, {\it Variable-basis topological systems versus variable-basis topological spaces}, Soft Comput., {\bf 14}\textbf{(10)} (2010), 1059-1068. \bibitem{Vickers1989} S.~Vickers, {\it Topology via logic}, Cambridge University Press, 1989. \bibitem{Vickers1993} S.~Vickers, {\it Information systems for continuous posets}, Theor. Comput. Sci., {\bf 114}\textbf{(2)} (1993), 201-229. \bibitem{Wen2008} G. F. Wen, F. G. Shi and H. Y. Li, {\it Almost $S^*$-compactness in $L$-topological spaces}, Iranian Journal of Fuzzy Systems, {\bf 5}\textbf{(3)} (2008), 31-44. \bibitem{Zadeh1965} L.~A. Zadeh, {\it Fuzzy sets}, Information and Control, {\bf 8} (1965), 338-365. | ||
آمار تعداد مشاهده مقاله: 2,435 تعداد دریافت فایل اصل مقاله: 1,467 |