On the Definition of Intuitionistic Fuzzy h-ideals of Hemirings

• Journal title : Kyungpook mathematical journal
• Volume 53, Issue 3,  2013, pp.435-457
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2013.53.3.435
Title & Authors
On the Definition of Intuitionistic Fuzzy h-ideals of Hemirings
Rahman, Saifur; Saikia, Helen Kumari;

Abstract
Using the Lukasiewicz 3-valued implication operator, the notion of an ($\small{{\alpha},{\beta}}$)-intuitionistic fuzzy left (right) $\small{h}$-ideal of a hemiring is introduced, where $\small{{\alpha},{\beta}{\in}\{{\in},q,{\in}{\wedge}q,{\in}{\vee}q\}}$. We define intuitionistic fuzzy left (right) $\small{h}$-ideal with thresholds ($\small{s,t}$) of a hemiring R and investigate their various properties. We characterize intuitionistic fuzzy left (right) $\small{h}$-ideal with thresholds ($\small{s,t}$) and ($\small{{\alpha},{\beta}}$)-intuitionistic fuzzy left (right) $\small{h}$-ideal of a hemiring R by its level sets. We establish that an intuitionistic fuzzy set A of a hemiring R is a ($\small{{\in},{\in}}$) (or ($\small{{\in},{\in}{\vee}q}$) or ($\small{{\in}{\wedge}q,{\in}}$)-intuitionistic fuzzy left (right) $\small{h}$-ideal of R if and only if A is an intuitionistic fuzzy left (right) $\small{h}$-ideal with thresholds (0, 1) (or (0, 0.5) or (0.5, 1)) of R respectively. It is also shown that A is a ($\small{{\in},{\in}}$) (or ($\small{{\in},{\in}{\vee}q}$) or ($\small{{\in}{\wedge}q,{\in}}$))-intuitionistic fuzzy left (right) $\small{h}$-ideal if and only if for any $\small{p{\in}}$ (0, 1] (or $\small{p{\in}}$ (0, 0.5] or $\small{p{\in}}$ (0.5, 1] ), $\small{A_p}$ is a fuzzy left (right) $\small{h}$-ideal. Finally, we prove that an intuitionistic fuzzy set A of a hemiring R is an intuitionistic fuzzy left (right) $\small{h}$-ideal with thresholds ($\small{s,t}$) of R if and only if for any $\small{p{\in}(s,t]}$, the cut set $\small{A_p}$ is a fuzzy left (right) $\small{h}$-ideal of R.
Keywords
Intuitionistic fuzzy set;Fuzzy h-ideal;Intuitionistic fuzzy ideal;Intuitionistic fuzzy h-ideal;Lukasiewicz implication operator;
Language
English
Cited by
References
1.
K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), 87-96.

2.
K. T. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems, 61(1994), 137-142.

3.
K. T. Atanassov, Intuitionistic fuzzy sets. Theory and applications, Studies in Fuzziness and Soft Computing, 35, Heidelberg, Physica-Verlag, 1999.

4.
S. K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems 51(1992), 235-241.

5.
S. K. Bhakat and P. Das, (${\in},\;{\in}\;{\vee}q$)-fuzzy subgroup, Fuzzy Sets and Systems, 80(1996), 359-368.

6.
S. K. Bhakat and P. Das, Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems, 81(1996), 383-393.

7.
Y. Bin and Y. Xue-hai, The normal intuitionistic fuzzy subgroups, Proceedings of 2010 IEEE International Conference on Intelligent Computing and Intelligent Systems (29-31, October, 2010, Xiamen ,China), 210-214.

8.
Y. Bin and Y. Xue-hai, Intuitionistic fuzzy subrings and ideas, 2011 8th international Conference on Fuzzy Systems and Knowledge Discovery, 1(2011), 300-305.

9.
B. Davvaz, J. M. Zhan and K. P. Shum, Generalized fuzzy $H_{v}$-submodules endowed with interval-valued membership functions, Information Sciences, 17 (2008), 3147-3157.

10.
B. Davvaz and P. Corsini, Redefined fuzzy $H_{v}$-submodules and many valued implications, Information Sciences, 177(2007), 865-875.

11.
W. A. Dudek, M. Shabir and M. I. Ali, (${\alpha},\;{\beta}$)- fuzzy ideal of hemirings, Computers and Mathematics with Applications, 58(2009), 310-321.

12.
W. A. Dudek, Special types of intuitionistic fuzzy left h-ideals of hemirings, Soft comput., 12(2008), 359-365.

13.
W. A. Dudek, Intuitionistic fuzzy h-ideals of hemirings, WSEAS Trans. Math., 12(2006), 1315-1321.

14.
U. Hebisch and H. J. Weinert, Semirings, Algebraic Theory and Applications in the Computer Science, World Scientific, 1998.

15.
H. Hossein, Generalized fuzzy k-ideals of semirings with interval-valued membership functions, Bull. Malays. Math. Sci. Soc., 32(3)(2009), 409-424.

16.
S. Jonathan and J. S. Golan, Semirings and their applications, Kluwer Academic Publishers, 1999.

17.
Y. B. Jun, M. A. Ozturk and S. Z. Song, On fuzzy h-ideals in hemirings, Information Sciences, 162(2004), 211-226.

18.
M. Kondo and W. A. Dudek, On the transfer principle in fuzzy theory, Mathware and Soft Computing, 12(2005), 41-55.

19.
J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, World scientific, 1998.

20.
S. Rahman, H. K. Saikia and B. Davvaz, On the definition of Atanassov's intuitionistic fuzzy subrings and ideals, Bull. Malays. Math. Sci. Soc., Accepted.

21.
A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35(1971), 512-517.

22.
X. H. Yuan, H. X Li and E. S. Lee, On the definition of the intuitionistic fuzzy subgroups, Computers and Mathematics with Applications, 59(2010), 3117-3129.

23.
X. H. Yuan, H. X. Li and K.B. Sun, The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval-valued fuzzy sets, Science in China Series F: Information Sciences, 39(2009), 933-945.

24.
L. A. Zadeh, Fuzzy sets, Inform. Control, 8(1965), 338-353.