Korean J. Math. Vol. 22 No. 2 (2014) pp.367-381
DOI: https://doi.org/10.11568/kjm.2014.22.2.367

Coincidences of different types of fuzzy ideals in ordered $\Gamma$-semigroups

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Published: Jun 26, 2014
Keywords:
fuzzy generalized bi-ideal, fuzzy bi-ideal, fuzzy interior ideal, fuzzy ideal and ordered $\Gamma$-semigroup
Mathematics Subject Classification:
06F99, 20N25

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Arunothai Kanlaya
Department of Mathematics, School of Science University of Phayao, Phayao 56000, THAILAND
Aiyared Iampan
University of Phayao

Abstract

The notion of $\Gamma$-semigroups was introduced by Sen in 1981 and that of fuzzy sets by Zadeh in 1965. Any semigroup can be reduced to a $\Gamma$-semigroup but a $\Gamma$-semigroup does not necessarily reduce to a semigroup. In this paper, we study the coincidences of fuzzy generalized bi-ideals, fuzzy bi-ideals, fuzzy interior ideals and fuzzy ideals in regular, left regular, right regular, intra-regular, semisimple ordered $\Gamma$-semigroups.



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Supporting Agencies

This research was supported by the Group for Young Algebraists in University of Phayao (GYA) Thailand.

References

[1] S. Abdullah, M. Aslam, B. Davvaz, and M. Naeem, A note on ordered semigroups characterized by their (∈,∈ ∨q)-fuzzy bi-ideals, UPB Scientific Bulletin, Series A 75 (2013), 41–44. Google Scholar

[2] S. Bashir, M. Amin, and M. Shabir, Prime fuzzy bi-ideals of Γ-semigroups, Ann. Fuzzy Math. Inform. 5 (2013), 115–128. Google Scholar

[3] R. Chinram and S. Malee, L-fuzzy ternary subsemirings and L-fuzzy ideals in ternary semirings, IAENG Int. J. Appl. Math. 40 (2010), IJAM 40 3 03. Google Scholar

[4] R. Chinram and S. Saelee, Fuzzy ideals and fuzzy filters of ordered ternary semi- groups, J. Math. Res. 2 (2010), 93–97. Google Scholar

[5] I. Chon, On fuzzy bi-ideals in semigroups, Korean J. Math. 19 (2011), 321–330. Google Scholar

[6] P. Dheena and S. Manivasan, Quasiideals of a P-regular near-rings, Internat. J. Google Scholar

[7] Algebra 5 (2011), 1005–1010. Google Scholar

[8] A. Iampan, Characterizing fuzzy sets in ordered Γ-semigroups, J. Math. Res. 2 Google Scholar

[9] (2010), 52–56. Google Scholar

[10] N. Kehayopulu and M. Tsingelis, A note on fuzzy sets in semigroups, Sci. Math. Google Scholar

[11] (1999), 411–413. Google Scholar

[12] N. Kehayopulu and M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Google Scholar

[13] Forum 65 (2002), 128–132. Google Scholar

[14] N. Kehayopulu and M. Tsingelis, Green’s relations in ordered groupoids in terms Google Scholar

[15] of fuzzy subsets, Soochow J. Math. 33 (2007), 383–397. Google Scholar

[16] F. M. Khan, N. H. Sarmin, and A. Khan, New types of generalized fuzzy bi Γ-ideals in ordered Γ-semigroups, Science International (Lahore) 25 (2013), 411–418. Google Scholar

[17] K. H. Kim, Intuitionistic fuzzy sets in ordered semigroups, International Math-ematical Forum 4 (2009), 2259–2268. Google Scholar

[18] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Pauli 27 (1979), 17–21. Google Scholar

[19] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5 (1981), 203–215. Google Scholar

[20] N. Kuroki, On fuzzy semigroups, Informing Science 53 (1991), 203–236. Google Scholar

[21] N. Kuroki, Fuzzy semiprime quasi-ideals in semigroups, Informing Science 75 (1993), 201–211. Google Scholar

[22] S. K. Majumder, On fuzzy weakly completely prime ideal in Γ-semigroups, SDU Journal of Science (E-Journal) 5 (2010), 230–238. Google Scholar

[23] S. K. Majumder and M. Mandal, Fuzzy generalized bi-ideals of Γ-semigroups, Fuzzy Information and Engineering 4 (2012), 389–399. Google Scholar

[24] S. K. Majumder and S. K. Sardar, On properties of fuzzy ideals in po-semigroup, Armen. J. Math. 2 (2009), 65–72. Google Scholar

[25] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512–517. Google Scholar

[26] S. Saelee and R. Chinram, A study on rough, fuzzy and rough fuzzy bi-ideals of ternary semigroups, IAENG Int. J. Appl. Math. 41 (2011), IJAM 41 3 01. Google Scholar

[27] N. K. Saha, On Γ-semigroup II, Bull. Calcutta Math. Soc. 79 (1987), 331–335. Google Scholar

[28] N. K. Saha, On Γ-semigroup III, Bull. Calcutta Math. Soc. 80 (1988), 1–12. Google Scholar

[29] S. K. Sardar, B. Davvaz, S. K. Majumder, and S. Kayal, On generalized fuzzy interior ideals in Γ-semigroups, Hacet. J. Math. Stat. 41 (2012), 231–241. Google Scholar

[30] S. K. Sardar, B. Davvaz, S. K. Majumder, and M. Mandal, Characteristic ideals and fuzzy haracteristic ideals of Γ-semigroups, Mathematica Aeterna 2 (2012), 189–201. Google Scholar

[31] M. K. Sen, On Γ-semigroups, Proceedings of the International conference on Algebra and its application. Decker Publication, New York, New York (1981), 301–308. Google Scholar

[32] M. K. Sen and N. K. Saha, On Γ-semigroup I, Bull. Calcutta Math. Soc. 78 (1986), no. 3, 180–186. Google Scholar

[33] M. K. Sen and A. Seth, On po-Γ-semigroups, Bull. Calcutta Math. Soc. 85 (1993), 445–450. Google Scholar

[34] M. Shabir and A. Khan, Fuzzy filters in ordered semigroups, Lobachevskii J. Math. 29 (2008), 82–89. Google Scholar

[35] T. Shah and I. Rehman, On Γ-ideals and Γ-bi-ideals in Γ-AG-groupoids, Inter- national Journal of Algebra 4 (2010), 267–276. Google Scholar

[36] D. R. Prince Williams, K. B. Latha, and E. Chandrasekeran, Fuzzy bi-Γ-ideals in Γ-semigroups, Hacet. J. Math. Stat. 38 (2009), 1–15. Google Scholar

[37] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338–353. Google Scholar

[38] J. M. Zhan and X. L. Ma, On fuzzy interior ideals in semigroups, J. Math. Res. Exposition 28 (2008), 103–110. Google Scholar