Document Type: Research Paper


1 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

2 Department of Mathematics Education and RINS, Gyeongsang National University, Chinju 660-701, Korea

3 Department of Mathematics, Air University E-9, PAF Complex, Islamabad, Pakistan


In this paper, we introduce and study the prime, strongly prime,
semiprime and irreducible fuzzy bi-ideals of a semigroup. We characterize those
semigroups for which each fuzzy bi-ideal is semiprime. We also characterize
those semigroups for which each fuzzy bi-ideal is strongly prime.


[1] J. Ahsan, R. M. Latif and M. Shabir, Fuzzy quasi-ideals in Semigroups, Journal of Fuzzy
Mathematics, 9 (2001), 259-270.
[2] J. Ahsan, K. Y. Li and M. Shabir, Semigroups characterized by their fuzzy bi-ideals, Journal
of Fuzzy Mathematics, 10 (2002), 441-449.
[3] J. Ahsan, K. Saifullah and M. F. Khan, Semigroups characterized by their fuzzy ideals, Fuzzy
Systems and Mathematics, 9 (1995), 29-32.
[4] J. Ahsan, K. Saifullah and M. Shabir, Fuzzy prime and semiprime S-subacts over monoids,
New Mathematics and Natural Computation, 3 (2007), 41-56.
[5] A. Bargiela and W. Pedrycz, Granular computing: an introduction, The Kluwer Inter. Series
in Engginearing and Computer Science, Kluwe Academic Publishers, Boston MA., ISBN
1-4020-7273-2, 717(xx) (2003), 452.

[6] G. Birkhoff, Lattice theory, Amer. Math. Soc., Coll. Publ., Providence, Rhode Island, 1967.
[7] N. Kehayopulu and M. Tsingelis, The embeding of an ordered groupoid into a poe-groupoid
in terms of fuzzy sets, Information Sciences, 152 (2003), 231-236.
[8] N. Kehayopulu and M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Information Sciences,
171 (2004), 13-28.
[9] N. Kehayopulu and M. Tsingelis, Regular ordered semigroups in terms of fuzzy subsets, Information
Sciences, 176 (2006), 3675-3693.
[10] G. J. Klir and B. Yuan, Fuzzy sets and fuzzy logic theory and applications, Prentice Hall Inc,
New Jersey, 1995.
[11] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Paul, 28 (1979), 17-21.
[12] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems, 5
(1981), 203-215.
[13] N. Kuroki, Fuzzy semiprime ideals in semigroups, Fuzzy Sets and Systems, 8 (1982), 71-79.
[14] N. Kuroki, On fuzzy semigroups, Information Sciences, 53 (1991), 203-236.
[15] S. Q. Li and Y. He, On semigroups whose bi-ideals are prime, Acta Mathematica Sinica, 49
(2006), 1189-1194.
[16] W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982),
[17] J. N. Mordeson and D. S. Malik, Fuzzy automata and languages, Theory and Applications,
Computational Mathematics Series, Chapman and Hall/CRC, Boca Raton, 2002.
[18] J. N. Mordeson, D. S. Malik and N. Kuroki, Fuzzy semigroups, Studies in Fuzziness and Soft
Computing, Springer-Verlag, Berlin, 131 (2003).
[19] W. Pedrycz and F. Gomide, An introduction to fuzzy sets: analysis and design, With a Foreword
by Lotfi A. Zadhe, Complex Adaptive Syst. A Bradford book, MIT Press, Cambridge,
MA, ISBN: 0-262-16171-0, xxiv (1998), 465.
[20] A. Rosenfeld, Fuzzy groups, Journal of Mathematical Analysis and Applications, 35 (1971),
[21] M. Shabir, Fully fuzzy prime semigroups, International Journal of Mathematics and Mathematical
Sciences, 1 (2005), 163-168.
[22] M. Shabir and Naila Kanwal, Prime bi-ideals in semigroups, Southeast Asian Bulletin of
Mathematics, 31 (2007), 757-764.
[23] E. Trillas, On the use of words and fuzzy sets, Information Sciences, 176 (2006), 1463-1487.
[24] D. Willaeys and N. Malvache, The use of fuzzy sets for the treatment of fuzzy information
by computer, Fuzzy Sets and System, 5 (1981), 323-328.
[25] X. Y. Xie, On prime fuzzy ideals of a semigroup, Journal of Fuzzy Mathematics, 8 (2000),
[26] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[27] L. A. Zadeh, Fuzzy sets and systems system theory (fox J. ed.), Microwave Research Institute
symposia series xv, Polytechnic Press Brook lyn, NY, (1965b), 29-37. Reprinted in Int. J. of
General Systems, 17 (1990), 129-138.
[28] L. A. Zadeh, Fuzzy sets and applications selected papers, Edited and with a Preface by R.
R. Yager, R. M. Tong, S. Ovchinnikov and H. T. Nguyen, A Wiley-Interscience Publication,
John Wiley and Sons Inc., New York, 1987.
[29] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1
(1987), 3-28.
[30] L. A. Zadeh, Fuzzy Sets, Fuzzy Logic and Fuzzy Systems Selected Papers by Lotfi A. Zadeh,
Edited and with a preface by George J. Klir and Bo Yuan, Advances in Fuzzy Systems-
Applications and Theory, World Scientific Publishing Co., 6 (1996).
[31] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)-an outline, Information
Sciences, 172 (2005), 1-40.
[32] H. J. Zimmermann, Fuzzy set theory and its applications, With a Foreword by L. A. Zadeh,
International Series in Management Science/Operation Research, Kluwer-Nijhoff Publishing,
Boston, 1985.

[33] H. J. Zimmermann, Fuzzy set theory and its applications, With a Foreword by L. A. Zadeh,
fourth edition, Kluwer Academic Publishers, Boston, 2001.