Open Access Open Access  Restricted Access Subscription or Fee Access

On Intuitionistic Fuzzy Ideals of Semirings with Respect to Fuzzy Connectives

Saifur Rahman, Apil Uddin Ahmed


Intuitionistic fuzzy prime and strongly irreducible ideals with respect to fuzzy connectives of a semiring are introduced and investigated some properties of these ideals. It is shown that if every cut of an intuitionistic fuzzy set of a semiring is an ideal, then the associated intuitionistic fuzzy set is both left and right ideal with respect to any t-norm , however, it is not necessarily an ideal with respect to . Finite intersection of intuitionistic fuzzy ideals (k-ideals) with respect to a t-norm is again is an intuitionistic fuzzy ideal (k-ideal) with respect to the t-norm is established.  It is also found that if an intuitionistic fuzzy set  is an ideal (k-ideal) with respect to the minimum t-norm, then its nonempty cut, strong cut and weak cut are ideals (k-ideals).


Intuitionistic Fuzzy Set, Semiring, t-norm, t-conorm, Strong α-cut, weak α-cut.

Full Text:



J. Ahsan, J. N. Mordeson and M. Shabir, “Fuzzy semirings with application to Automata theory”, Springer, vol.278, 2012, pp. 18-43.

J.Ahsan, J. Saifullah and K. Khan, “Fuzzy semirings”, Fuzzy Set and Systems, vol.60, 1993, pp. 302-309.

J. Ahsan, J. Saifullah and M. Shabir, “Fuzzy prime ideals of a semiring and Fuzzy prime subsemimodules of semi modules over a semiring”, New Mathematics and Natural computation, vol.2, no. 3, 2006, pp. 219-236.

M. Akram, and W. A. Dudek, “Intuitionistic fuzzy left k-ideals of semirings”, Soft Computation, vol.12, 2008, pp.881-890.

K. T. Atanassov, “Intuitionistic fuzzy sets”, Seventh Scientific Session of ITKR, Sofia, June 1983.

K. T. Atanassov, “Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, vol.20, 1986, pp. 87-96.

R. Biswas, “Intuitionistic fuzzy subgroups”, Math. Forum, vol.10, 1989, pp. 37-46.

G. Deschrijver, “On the representation of intuitionistic fuzzy t-norms and t-conorms”, IEEE Trans. of Fuzzy Systems, vol. 12, no. 1, pp. 45-61, 2004.

G. Deschrijver, “The Archimedean property for t-norms in interval-valued fuzzy set theory”, Fuzzy Sets and Systems, vol.157, 2006, pp.2311-2327.

V. Janis, “t-norm based cuts of intuitionistic fuzzy set”, Information Sciences, vol.180, 2010, pp. 1134-1137.

S. Jenei, “New family of triangular norms via contrapositive symmetrization of residuated implications”, Fuzzy Sets and Systems, vol.110, 2000, pp.157-174.

G. J. Klirand B. Yuan, “Fuzzy sets and Fuzzy logic theory and Applications”, Prentice Hall of India, 2008.

J. N. Mordesonand D. S. Malik, “Fuzzy Commutative Algebra”, World Scientific Publishing, 1998.

J. N. Mordeson, K. R. Bhutani and A. Rosenfeld, “Fuzzy Group Theory”, Springer, 2005.

E. Palmeira, B. Bedregal, R. Mesiar, and J. Fernandez, “A new way to extend t-norms, t-conorms and negations”, Fuzzy Sets and Systems, vol. 240, 2014, pp.1-21.

A. Pankowska and M. Wygralak, “General IF-sets with triangular norms and their applications to group decision making”, Information Sciences vol.176, 2006, pp. 2713-2754.

S. G.Pushkov, “Fuzzy modules with respect to a t-norm and some of their properties”, Journal of Mathematical Sciences”, vol.154, no. 3, 2008, pp. 374-378.

S.R.ahman, “On cuts of Atanassov’s intuitionistic fuzzy sets with respect to fuzzy connectives”, Information Scinces, vol.340-341, 2016, pp.262-278.

S. Rahman and H. K. Saikia,“Some aspects of Atanassov’s intuitionistic fuzzy submodule”, International Journal of Pure and Applied mathematics, vol.77, no. 3, 2012, pp. 369-383.

S. Rahman,H. K. Saikiaand B. Davvaz, “On the definition of Atanassov’s intuitionistic fuzzy subrings and ideals”, Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, no. 2, 2013, pp. 401–418.

S. Rahman and H. K. Saikia, “Atanassov’s intuitionistic fuzzy submodules with respect to a t – norm”, Soft Comput., vol. 17, no. 7, 2013, pp. 1253-1262.

H. S. Vandiver, “Note on a simple type of algebra in which the cancellation law of addition not hold”, Bulletin of the American Mathematical Sciences Society, vol. 40, no. 12, 1934, pp. 916-920.

L. A. Zadeh, “Fuzzy sets”, Inform. Control, vol. 8, 1965, pp. 338-353.


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.