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

Arunothai Kanlaya, Aiyared Iampan

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.


Keywords


fuzzy generalized bi-ideal, fuzzy bi-ideal, fuzzy interior ideal, fuzzy ideal and ordered $\Gamma$-semigroup

Subject classification

06F99; 20N25

Sponsor(s)

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

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References


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)

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

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)

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

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

P. Dheena and S. Manivasan, Quasiideals of a P-regular near-rings, Internat. J. (Google Scholar)

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

A. Iampan, Characterizing fuzzy sets in ordered Γ-semigroups, J. Math. Res. 2 (Google Scholar)

(2010), 52–56. (Google Scholar)

N. Kehayopulu and M. Tsingelis, A note on fuzzy sets in semigroups, Sci. Math. (Google Scholar)

(1999), 411–413. (Google Scholar)

N. Kehayopulu and M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup (Google Scholar)

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

N. Kehayopulu and M. Tsingelis, Green’s relations in ordered groupoids in terms (Google Scholar)

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

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)

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

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

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

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

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

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

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

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

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

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)

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

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

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)

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)

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)

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

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

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

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)

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)

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

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


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