# ROUGH PRIME IDEALS AND ROUGH FUZZY PRIME IDEALS IN GAMMA-SEMIGROUPS

Chinram, Ronnason

• Published : 2009.07.31
• 77 50

#### Abstract

The notion of rough sets was introduced by Z. Pawlak in the year 1982. The notion of a $\Gamma$-semigroup was introduced by M. K. Sen in the year 1981. In 2003, Y. B. Jun studied the roughness of sub$\Gamma$-semigroups, ideals and bi-ideals in i-semigroups. In this paper, we study rough prime ideals and rough fuzzy prime ideals in $\Gamma$-semigroups.

#### Keywords

congruences;${\Gamma}$-semigroups;quotient ${\Gamma}$-semigroups;prime ideals;$\theta$-lower rough prime ideals;$\theta$-upper rough prime ideals

#### References

1. M. Banerjee and S. K. Pal, Roughness of a fuzzy set, Information Science 93 (1996), no. 3-4, 235–246 https://doi.org/10.1016/0020-0255(96)00081-3
2. R. Biswas and S. Nanda, Rough groups and rough subgroups, Bulletin of the Polish Academy of Sciences, Mathematics 42 (1994), no. 3, 251–254
3. R. Chinram, On quasi-gamma-ideals in gamma-semigroups, ScienceAsia 32 (2006), no. 4, 351–353
4. R. Chinram and C. Jirojkul, On bi-$\Gamma$-ideals in $\Gamma$-semigroups, Songklanakarin Journal of Science and Technology 29 (2007), no. 1, 231–234
5. B. Davvaz, Rough subpolygroups in a factor polygroup, Journal of Intelligent and Fuzzy Systems 17 (2006), no. 6, 613–621
6. J. Iwinski, Algebraic approach to rough sets, Bulletin of the Polish Academy of Sciences, Mathematics 35 (1987), 673–683
7. Y. B. Jun, Roughness of gamma-subsemigroups/ideals in gamma-semigroups, Bull. Korean Math. Soc. 40 (2003), no. 3, 531–536 https://doi.org/10.4134/BKMS.2003.40.3.531
8. N. Kuroki, Fuzzy congruences and fuzzy normal subgroups, Information Sciences 60 (1992), no. 3, 247–259 https://doi.org/10.1016/0020-0255(92)90013-X
9. N. Kuroki, Rough ideals in semigroups, Information Science 100 (1997), no. 1-4, 139–163 https://doi.org/10.1016/S0020-0255(96)00274-5
10. Y. I. Kwon and S. K. Lee, On weakly prime ideals of ordered $\Gamma$-semigroups, Commun. Korean Math. Soc. 13 (1998), no. 2, 251–256
11. N. K. Saha, On $\Gamma$-semigroup II, Bull. Calcutta Math. Soc. 79 (1987), no. 6, 331–335
12. N. K. Saha, On $\Gamma$-semigroup III, Bull. Calcutta Math. Soc. 80 (1988), no. 1, 1–12
13. M. K. Sen, On $\Gamma$-semigroup, Algebra and its applications (New Delhi, 1981), 301–308, Lecture Note in Pure and Appl. Math., 91, Dekker, New York, 1984
14. M. K. Sen and N. K. Saha, On $\Gamma$-semigroup I, Bull. Calcutta Math. Soc. 78 (1986), 181–186
15. M. Siripitukdet and A. Iampan, On the ordered n-prime ideals in ordered $\Gamma$-semigroups, Commun. Korean Math. Soc. 23 (2008), no. 1, 19–27 https://doi.org/10.4134/CKMS.2008.23.1.019
16. R. Chinram and P. Siammai, On green's relations for $\Gamma$-semigroups and reductive $\Gamma$-semigroups, International Journal of Algebra 2 (2008), no. 4, 187–195
17. Y. B. Jun, On closure gamma-semigroups, Commun. Korean Math. Soc. 19 (2004), no. 4, 639–641 https://doi.org/10.4134/CKMS.2004.19.4.639
18. N. Kuroki and P. P. Wang, The lower and upper approximations in a fuzzy group, Information Science 90 (1996), no. 1-4, 203–220 https://doi.org/10.1016/0020-0255(95)00282-0
19. Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (1982), 341–356 https://doi.org/10.1007/BF01001956
20. Q. M. Xiao and Z. L. Zhang, Rough prime ideals and rough fuzzy prime ideals in semigroups, Information Sciences 176 (2006), no. 6, 725–733 https://doi.org/10.1016/j.ins.2004.12.010

#### Cited by

1. On fuzzy interior Γ-hyperideals in ordered Γ-semihypergroups vol.32, pp.3, 2017, https://doi.org/10.3233/JIFS-16431
2. A study on (fuzzy) quasi-Γ-hyperideals in ordered Γ-semihypergroups vol.32, pp.6, 2017, https://doi.org/10.3233/IFS-162117
3. Applications of rough sets to Γ-hyperideals in left almost Γ-semihypergroups vol.21, pp.S1, 2012, https://doi.org/10.1007/s00521-012-0809-5
4. Rough Fuzzy Hyperideals in Ternary Semihypergroups vol.2012, 2012, https://doi.org/10.1155/2012/595687
5. Approximations of bipolar fuzzy $$\Gamma$$Γ-hyperideals of $$\Gamma$$Γ-semihypergroups vol.29, pp.5-6, 2018, https://doi.org/10.1007/s13370-018-0585-1