AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC

Document Type: Research Paper

Author

Department of Mathematics, Faculty of Mathematics and Com- puter, Shahid Bahonar University of Kerman, Kerman, Iran

Abstract

In this paper we extend the notion of  degrees of membership and non-membership of intuitionistic fuzzy sets to lattices and  introduce a residuated lattice with appropriate operations to serve as semantics of intuitionistic fuzzy logic. It would be a step forward to find an algebraic counterpart for intuitionistic fuzzy logic. We give the main properties of the operations defined and prove some theorems to demonstrate our goal.

Keywords


bibitem{1 }   K. T. Atanassov, {it Intuitionistic fuzzy sets}, Fuzzy Sets and Systems, {bf20} (1986), 87-96.

bibitem{2 }   K. T. Atanassov and S. Stoeva, {it Intuitionistic L - fuzzy sets}, In: R. Trappl, ed., Elsevier Science Publishers B.V., Noth Holland, 1984.

bibitem{3 } K. T. Atanassov and G. Gargov, {it Elements of intuitionistic fuzzy logic. part I}, Fuzzy Sets and Systems, {bf95} (1998), 39-52.


bibitem{4 } M. Baczynski, {it Residual implications revisited}, Fuzzy Sets and Systems, {bf 145} (2004), 267- 277.

bibitem{5}  P. Burillo and H. Bustince, {it Intuitionistic fuzzy relations. effects of Atanassov's operators on the properties of intuitionistic fuzzy relations}, Mathware & Soft Computing, {bf2} (1995), 117- 148.

 bibitem{6}  R. Cignoli and F. Esteva, {it Commutative integral bounded residuated lattices with an added involution}, Annals of Pure and Applied Logic, {bf161} (2009), 150-160.

  bibitem{7}   P. Cintula, {it From fuzzy logic to fuzzy mathematics}, Ph.D. Thesis, Technical University, Prague, 2005.

  bibitem{8}   C. Cornelis, G. Deschrijver and E. E. Kerre, {it Classification on intuitionistic fuzzy implicators: an algebraic approach}, In Proceedings of the FT & T' 02, Durham, North Carolina, 105-108.
 

bibitem{9}   G. Deschrijver, C. Cornelis and E. E. Kerre, {it Intuitionistic fuzzy connectives revisited}, In Proceedings of IPMU'02, July 1-5, 2002.
 

 bibitem{10}  J. A. Goguen, {it L - Fuzzy sets}, Journal of Math. Anal. And Applications, {bf18} (1967), 145-173.

  bibitem{11}  P. Hajek, {it Metamathematics of fuzzy logic}, Trends in Logic, Kluwer Acad.Publ., Drdrecht, {bf4} (1998).

  bibitem{12}  P. Hajek, {it What is mathematical fuzzy logic?}, Fuzzy Sets and Systems, {bf157} ( 2006), 597-603.

  bibitem{13}  Y. Hong, X. Ruiping and  F. Xianwen, {it Characterizing ordered semigroups by means of intuitionistic fuzzy bi- ideals}, Mathware & Soft Computing, {bf14} (2007), 57-66.

 bibitem{14}   H. Ono, {it Subsructural logics and ResiduatedLattices-an introduction}, Trends in Logic, {bf20} (2003), 177-212.

 bibitem{15}   P. Smets and P. Magrez, {it Implications in fuzzy logic}, Int. J. of Approximate Reasoning, {bf1} (1987), 327-347.

 bibitem{16}   E. Szmidt and J. Kacprzyk, {it Intuitiinistic fuzzy sets in some medical applications, computational intelligence}, Theory and Applications, Lecture Notes in Computer Science, (2001), V. 2206/2001, 148-151.

 bibitem{17}    E. Szmidt and K. Marta, {it Atanassov's intuitionistic fuzzy sets in classification of imbalanced and overlapping classes}, Studies in Computational Intelligence (SCI), {bf109} (2008), 455- 471.

 bibitem{18}A. Tepavcevic and M. G. Ranitovic, {it General form of lattice valued intuitionistic fuzzy sets}, Computational Intelligence, Theory and Applications, {bf14} (2006), 375-381.
 bibitem{19}A. Tepavcevic and T. Gerstenkorn, {it Lattice valued intuitionistic fuzzy sets}, Central European Journal of Mathematics, {bf2(3)} (2004), 388-398.
 

bibitem{20} E. Turunen, {it Mathematics behind fuzzy logic}, Advances in Soft Computing, Physica-Verlag, Heidelberg, 1999.
 

bibitem{21}  I. K. Vlachos and G. D. Sergiadis, {it Towards Intuitionistic fuzzy image processing}, Proceedings of the 2005 International Conference on Computational Intelligence for Modelling, Control and Automation.
   

  bibitem{22}M. Ward  and R. P. Dilworth,  {it"Residuated lattices", Trans. Amer. Math. Soc.}, {bf45} (1939), 335-54, Reprinted in Bogart, K, Freese, R., and Kung, J., eds., 1990.
       

  bibitem{23}   L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf8(3)} (1965), 338-353.
        

 bibitem{24}   L. A. Zadeh, {it Fuzzy sets, fuzzy logic and fuzzy systems}, Selected papers by Lotfi A. Zadeh, Editors George J. Klir and Bo Yuan, World Scientific, 1996.