Fuzzy semi-quantales, (L,M) quasi-fuzzy topological spaces and their duality

The present paper introduces M-fuzzy semi-quantales, fuzzifying semi-quantales, and (L, M)-quasi-fuzzy topological spaces, providing a common framework for (L, M)-fuzzy topological spaces of Kubiak and Šostak, L-quasi-fuzzy topological spaces of Rodabaugh and L-fuzzy topological spaces of Höhle and Šostak. In this paper, we set up a dual adjunction between the category of (L, M)-quasi-fuzzy topological spaces and the category of M-fuzzy semi-quantales, and then show that this adjunction includes a dual equivalence between the category of (L, M)-sober (L, M)-quasi-fuzzy topological spaces and the category of (L, M)-spatial M-fuzzy semi-quantales.

[1]  David M. Clark,et al.  Natural Dualities for the Working Algebraist , 1998 .

[2]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[3]  M. Stone.  The theory of representations for Boolean algebras , 1936 .

[4]  Horst Herrlich,et al.  Abstract and concrete categories , 1990 .

[5]  Alexander P. Sostak,et al.  Axiomatic Foundations Of Fixed-Basis Fuzzy Topology , 1999 .

[6]  John R. Isbell,et al.  Atomless Parts of Spaces. , 1972 .

[7]  Marcel Erné,et al.  General Stone duality , 2004 .

[8]  Mustafa Demirci,et al.  Fundamental duality of abstract categories and its applications , 2014, Fuzzy Sets Syst..

[9]  Stephen Ernest Rodabaugh,et al.  Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics , 2007, Int. J. Math. Math. Sci..

[10]  Christian Eitzinger,et al.  Triangular Norms , 2001, Künstliche Intell..

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

[12]  E. M. Hartwell.  Boston , 1906 .

[13]  Mustafa Demirci,et al.  Pointed semi-quantales and lattice-valued topological spaces , 2010, Fuzzy Sets Syst..

[14]  A. Sostak,et al.  Foundations of the theory of (L, M)-fuzzy topological spaces , 2008 .

[15]  Radim Bělohlávek,et al.  Fuzzy Relational Systems: Foundations and Principles , 2002 .

[16]  Dona Papert,et al.  Sur les treillis des ouverts et les paratopologies , 1958 .

[17]  Wei Yao,et al.  A survey of fuzzifications of frames, the Papert-Papert-Isbell adjunction and sobriety , 2012, Fuzzy Sets Syst..

[18]  Maria Adler,et al.  Mathematical Principles Of Fuzzy Logic , 2016 .

[19]  K. I. Rosenthal.  Quantales and their applications , 1990 .

[20]  Sergey A. Solovyov,et al.  Sobriety and spatiality in varieties of algebras , 2008, Fuzzy Sets Syst..