# K-FLAT PROJECTIVE FUZZY QUANTALES

Document Type : Research Paper

Authors

1 College of Mathematics and Information Science, Shaanxi Normal Univer- sity, Xi'an 710119, P.R. China

2 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, P.R. China

Abstract

In this paper, we introduce the notion of {\bf K}-flat projective fuzzy quantales, and give an elementary characterization in terms of a fuzzy binary relation on the fuzzy quantale. Moreover, we  prove that {\bf K}-flat projective fuzzy quantales are precisely the coalgebras for a certain comonad on the category of fuzzy quantales. Finally, we present two special cases of {\bf K} as examples.

Keywords

#### References

[1] J. Adamek and H. Herrlich and G. E. Strecker, Abstract and Concrete Categories: The Joy
of Cats, John Wiley & Sons, New York, (1990), 1-507.
[2] B. Banaschewski, Projective frames: a general view, Cahiers Topologie Geom. Di erentielle
Cat., XLVI (2005), 301-312.
[3] R. Belohlavek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Aca-
demic/Plenum Publishers, New York, 20 (2002), 1-369.
[4] R. P. Dilworth, Non-commutative residuated lattices, Trans. Amer. Math. Soc., 46 (1939),
426-444.
[5] L. Fan, A new approach to quantitative domain theory, Electron. Notes Theor. Comput. Sci.,
45(1) (2001), 77-87.
[6] G. Gierz, et al., Continuous Lattices and Domains, Encyclopedia of Mathematics and its
Applications, vol. 93, Cambridge University Press, Cambridge, 93 (2003), 1-591.
[7] H. Herrlich and G. E. Strecker, Category Theory, An introduction, Second edition, Sigma
Series in Pure Mathematics, vol. 1, Heldermann Verlag, Berlin, 1 (1979), 1-400.
[8] P. T. Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics, Cambridge
University Press, Cambridge, 3 (1982), 1-370.
[9] D. Kruml and J. Paseka, Algebraic and categorical ascepts of quantales, Handb. Algebra, 5
(2008), 323-362.
[10] H. L. Lai and D. X. Zhang, Complete and directed complete
-categories, Theor. Comput.
Sci., 388 (2007), 1-25.
[11] Y. M. Li, M. Zhou and Z. H. Li, Projective and injective objects in the category of quantales,
J. Pure Appl. Algebra, 176 (2002), 249-258.
[12] J. Lu and B. Zhao, The projective objects in the category of fuzzy quantales, J. Shandong
Univ. (Nat. Sci.), (in Chinese), 50(2) (2015), 47-54 .
[13] C. J. Mulvey, &, Supplemento ai Rendiconti del Circolo Matematico di Palermo, II(12) (19
86), 99-104.
[14] C. J. Mulvey and J. W. Pelletier, On the quantisation of points, J. Pure Appl. Algebra, 159
(2001), 231-295.
[15] J. Paseka, Projective quantale: A general view, Int. J. Theor. Phys., 47(1) (2008), 291-296.
[16] K. I. Rosenthal, Quantales and their Applications, Pitman Research Notes in Mathematics
Series, vol. 234, Longman Scienti c & Technical, Essex, 234 (1990), 1-165.
[17] S. A. Solovyov, A representation theorem for quantale algebras, Contrib. Gen. Algebra, 18
(2008), 189-198.
[18] K. Y. Wang and B. Zhao, Some properties of the category of fuzzy quantales, J. Shaanxi
Norm. Univ. (Nat. Sci. Ed.), (in Chinese), 41(3) (2013), 1-6 .
[19] K. Y. Wang, Some researches on fuzzy domains and fuzzy quantales, Ph. D. Thesis, College
of Mathematics and Information Science, Shaanxi Normal University, Xi'an, (2012), 1-115.
[20] M. Ward, Structure residuation, Ann. Math., 39 (1938), 558-568.
[21] R. Wang and B. Zhao, Quantale algebra and its algebraic ideal, Fuzzy Syst. Math., (in
Chinese), 24 (2010), 44-49.
[22] M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc., 45 (1939), 335-
354.
[23] W. Yao and L. X. Lu, Fuzzy Galois connections on fuzzy posets, Math. Log. Quart., 55(1)
(2009), 105-112.
[24] W. Yao, Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete
posets, Fuzzy Sets Syst., 161(7) (2010), 973-987.
[25] W. Yao, An approach to fuzzy frames via fuzzy posets, Fuzzy Sets Syst., 166 (2011), 75-89.
[26] W. Yao, A survey of fuzzi cations of frames, the Papert-Papert-Isbell adjunction and sobri-
ety, Fuzzy Sets Syst., 190 (2012), 63-81.
[27] L. A. Zadeh, Fuzzy sets, Inf. Control, 8(3) (1965), 338-353.
[28] Q. Y. Zhang and L. Fan, Continuity in quantitative domains, Fuzzy Sets Syst., 154(1)
(2005), 118-131.