Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their Duality

Mustafa Demirci

2015

Abstract

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 Sostak, ˇ L-quasi-fuzzy topological spaces of Rodabaugh and L-fuzzy topological spaces of Hohle and ¨ Sostak. In this ˇ 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.

References

  1. Adámek, J., Herrlich, H. and Strecker, G. E. (1990). Abstract and Concrete Categories. New York: John Wiley & Sons..
  2. Be?lohlávek, R. (2002). Fuzzy Relational Systems. New York: Kluwer Academic Publishers.
  3. Clark, D. M. and Davey, B. A. (1998). Natural Dualities for the Working Algebraist. Cambrige: Cambridge University Press.
  4. Demirci, M. (2010). Pointed Semi-Quantales and LatticeValued Topological Spaces. Fuzzy Sets and Systems, 161, 1224-1241.
  5. Demirci, M. (2014). Fundamental Duality of Abstract Categories and Its Applications. Fuzzy Sets and Systems, 256, 73-94.
  6. Erné, M. (2004). General Stone duality. Topology and Its Applications, 137, 125-158.
  7. Hájek, P. (1998). Metamathematics of Fuzzy Logics. Dordrecht: Kluwer Academic Publishers.
  8. Höhle, U. (2001). Many Valued Topology and Its Applications. Boston: Kluwer Academic Publishers.
  9. Höhle, U. and S?ostak, A. P. (1999). Axiomatic Foundations of Fixed-Basis Fuzzy Topology. In Höhle, U. and Rodabaugh, S. E. (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (pp.123-272). Boston: Kluwer Academic Publishers.
  10. Isbell, J. R. (1972). Atomless Parts of Spaces. Mathematica Scandinavica, 31, 5-32.
  11. Johnstone, P.T. (1986). Stone Spaces. Cambridge: Cambridge University Press.
  12. Klement, E. P., Mesiar, R. and Pap, E. (2000). Triangular Norms. Dordrecht: Kluwer Academic Publishers.
  13. Kubiak, T. and S?ostak, A. (2009). Foundations of the Theory of (L,M)-fuzzy Topological Spaces. In Bodenhofer, U., DeBaets, B., Klement, E. P. and SamingerPlatz, S. (Eds.), Abstracts of the 30th Linz Seminar on Fuzzy Set Theory (pp. 70-73). Linz: Johannes Kepler Universität.
  14. Lawson, J. D. (1979). The Duality of Continuous Posets. Houston Journal of Mathematics, 5, 357-386. Math. 5 (1979) 357-386.
  15. Novák, V., Perfilieva, I. and Moc?ko?r, J. (1999). Mathematical Principles of Fuzzy Logic. Dordrecht: Kluwer Academic Publishers.
  16. Papert, D. and Papert, S. (1957/1958). Sur Les Treillis Des Ouverts Et Les Paratopologies. Seminaire Ehresmann. Topologie et Geometrie Differentielle, 1, 1-9.
  17. Porst, H. E. and Tholen, W. (1990). Concrete Dualities. In Herrlich, H. and Porst, H. E. (Eds.), Category Theory at Work (pp. 111-136). Berlin: Heldermann Verlag.
  18. Rodabaugh, S. E. (2007). Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics. International Journal of Mathematics and Mathematical Sciences, 71 pages. doi:10.1155/2007/43645.
  19. Rosenthal, K. I. (1990). Quantales and Their Applications. New York: Longman Scientific and Technical.
  20. Solovyov, S. A. (2008). Sobriety and Spatiality in Varieties of Algebras. Fuzzy Sets and Systems, 159, 2567-2585.
  21. Stone, M. H. (1936). The Theory of Representations for Boolean Algebras. Transactions of the American Mathematical Society, 40, 37-111.
  22. Yao, W. (2012). A Survey of Fuzzifications of Frames, the Papert-Papert-Isbell Adjunction and Sobriety. Fuzzy Sets and Systems, 190, 63-81.
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Paper Citation


in Harvard Style

Demirci M. (2015). Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their Duality . In Proceedings of the 7th International Joint Conference on Computational Intelligence - Volume 1: FCTA, (ECTA 2015) ISBN 978-989-758-157-1, pages 105-111. DOI: 10.5220/0005583601050111


in Bibtex Style

@conference{fcta15,
author={Mustafa Demirci},
title={Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their Duality},
booktitle={Proceedings of the 7th International Joint Conference on Computational Intelligence - Volume 1: FCTA, (ECTA 2015)},
year={2015},
pages={105-111},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005583601050111},
isbn={978-989-758-157-1},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 7th International Joint Conference on Computational Intelligence - Volume 1: FCTA, (ECTA 2015)
TI - Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their Duality
SN - 978-989-758-157-1
AU - Demirci M.
PY - 2015
SP - 105
EP - 111
DO - 10.5220/0005583601050111