Fuzzy sets and sheaves. Part II: : Sheaf-theoretic foundations of fuzzy set theory with applications to algebra and topology

This essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory, resp. of the theory of complete Ω-valued sets. Hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect. Part II of this series of two papers explains the sheaf-theoretic basis of Zadeh's operations on fuzzy sets (with the exception of the complementation). Further, the quotient problem w.r.t. similarity relations and the quotient problem of fuzzy groups w.r.t. fuzzy congruence relations are solved. Finally, stratified fuzzy topologies are identified with internal topologies on constant sheaves.

[1]  J. M Anthony,et al.  Fuzzy groups redefined , 1979 .

[2]  S. Ovchinnikov An Introduction to Fuzzy Relations , 2000 .

[3]  U. Höhle Presheaves over GL-monoids , 1995 .

[4]  U. Höhle M-valued Sets and Sheaves over Integral Commutative CL-Monoids , 1992 .

[5]  U. Höhle Many Valued Topology and its Applications , 2001 .

[6]  Oswald Wyler,et al.  Lecture notes on Topoi and Quasitopoi , 1991 .

[7]  L. N. Stout Topological space objects in a topos II: ɛ-Completeness and ɛ-cocompleteness , 1975 .

[8]  S. Gottwald A Treatise on Many-Valued Logics , 2001 .

[9]  Ulrich Höhle,et al.  Partial ordering in L-underdeterminate sets , 1985, Inf. Sci..

[10]  Lebesgue theory on a Boolean algebra , 1942 .

[11]  Joan Wick Pelletier,et al.  On the quantisation of points , 2001 .

[12]  Stephen E. Rodabaugh,et al.  The Hausdorff separation axiom for fuzzy topological spaces , 1980 .

[13]  Horst Herrlich,et al.  Category theory , 1979 .

[14]  Ju Pil Kim,et al.  Fuzzy congruences in groups , 1997, Fuzzy Sets Syst..

[15]  D. Scott,et al.  Sheaves and logic , 1979 .

[16]  Orrin Frink,et al.  New Algebras of Logic , 1938 .

[17]  Ulrich Bodenhofer,et al.  A Similarity-Based Generalization of Fuzzy Orderings Preserving the Classical Axioms , 2000, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[18]  I. J. Kumar,et al.  Fuzzy normal subgroups and fuzzy quotients , 1992 .

[19]  Etienne Kerre,et al.  The cutting of compositions , 1994 .

[20]  Ulrich Höhle,et al.  Fuzzy topologies and topological space objects in a topos , 1986 .

[21]  Dan A. Ralescu,et al.  REPRESENTATION THEOREMS FOR FUZZY CONCEPTS , 1975 .

[22]  M. Menger,et al.  Ensembles flous et fonctions aléatoires , 2003 .

[23]  Lotfi A. Zadeh,et al.  Similarity relations and fuzzy orderings , 1971, Inf. Sci..

[24]  U. Höhle On the Fundamentals of Fuzzy Set Theory , 1996 .

[25]  Venkat Murali,et al.  Fuzzy congruence relations , 1991 .

[26]  D. Dubois,et al.  Fuzzy Sets: History and Basic Notions , 2000 .

[27]  M. K. Chakraborty,et al.  Reduction of fuzzy strict order relations , 1985 .

[28]  Ulrich Höhle,et al.  A note on the hypergraph functor , 2002, Fuzzy Sets Syst..

[29]  Radko Mesiar,et al.  Fuzzy Interval Analysis , 2000 .

[30]  Ulrich Höhle,et al.  Separated Presheaves of Normed Spaces and -valued Norms , 2003, Int. J. Gen. Syst..

[31]  R. Goldblatt Topoi, the Categorial Analysis of Logic , 1979 .

[32]  Vladik Kreinovich,et al.  Methodology of Fuzzy Control: An Introduction , 1998 .

[33]  L. P. Belluce,et al.  Semi-simple and complete MV-algebras , 1992 .

[34]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[35]  R. Lowen Fuzzy topological spaces and fuzzy compactness , 1976 .

[36]  J. Recasens,et al.  Fuzzy Equivalence Relations: Advanced Material , 2000 .

[37]  Prabir Bhattacharya,et al.  Fuzzy normal subgroups and fuzzy cosets , 1984, Inf. Sci..

[38]  M. Garzon,et al.  Free fuzzy groups and fuzzy group presentations , 1992 .

[39]  L. N. Stout,et al.  Foundations of fuzzy sets , 1991 .

[40]  Sukhamay Kundu Membership functions for a fuzzy group from similarity relations , 1999, Fuzzy Sets Syst..

[41]  Wen-Xiang Gu,et al.  Generating fuzzy factor groups and the fundamental theorem of isomorphism , 1996, Fuzzy Sets Syst..

[42]  Gu Wen-xiang,et al.  Product structure of the fuzzy factor groups , 1993 .

[43]  C. L. Chang,et al.  Fuzzy topological spaces , 1968 .

[44]  A. Sostak On a fuzzy topological structure , 1985 .

[45]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[46]  P. Das Fuzzy groups and level subgroups , 1981 .

[47]  Pu Pao-Ming,et al.  Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence , 1980 .

[48]  Ulrich Höhle,et al.  Topological aspects of non-convergent sequences--a comment on Burgin's concept of fuzzy limits , 2005, Fuzzy Sets Syst..

[49]  John W. Gray,et al.  Sheaves with values in a category , 1965 .