Characterizations of Ordered Semigroups by (∈,∈ ∨q)-Fuzzy Ideals
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32847
Characterizations of Ordered Semigroups by (∈,∈ ∨q)-Fuzzy Ideals

Authors: Jian Tang

Abstract:

Let S be an ordered semigroup. In this paper we first introduce the concepts of (∈,∈ ∨q)-fuzzy ideals, (∈,∈ ∨q)-fuzzy bi-ideals and (∈,∈ ∨q)-fuzzy generalized bi-ideals of an ordered semigroup S, and investigate their related properties. Furthermore, we also define the upper and lower parts of fuzzy subsets of an ordered semigroup S, and investigate the properties of (∈,∈ ∨q)-fuzzy ideals of S. Finally, characterizations of regular ordered semigroups and intra-regular ordered semigroups by means of the lower part of (∈ ,∈ ∨q)-fuzzy left ideals, (∈,∈ ∨q)-fuzzy right ideals and (∈,∈ ∨q)- fuzzy (generalized) bi-ideals are given.

Keywords: Ordered semigroup, regular ordered semigroup, intraregular ordered semigroup, (∈, ∈ ∨q)-fuzzy left (right) ideal of an ordered semigroup, (∈, ∈ ∨q)-fuzzy (generalized) bi-ideal of an ordered semigroup.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1080702

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1972

References:


[1] S. K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems, 51(2) (1992), 235-241.
[2] S. K. Bhakat and P. Das, (Ôêê,Ôêê Ôê¿q)-fuzzy subgroup, Fuzzy Sets and Systems, 80(3) (1996), 359-368.
[3] B. Davvaz, (Ôêê,Ôêê Ôê¿q)-fuzzy subnearrings and ideals, Soft. Comput., 10 (2006), 206-211.
[4] Y. B. Jun and S. Z. Song, Generalized fuzzy interior ideals in semigroups, Inform. Sci., 176 (2006), 3079-3093.
[5] O. Kazanc─▒ and S. Yamak, Generalized fuzzy bi-ideals of semigroup, Soft. Comput., 12 (2008), 1119-1124.
[6] X. Ma and J. Zhan, Generalized fuzzy h-bi-ideals and h-quasi-ideals of hemirings, Inform. Sci., 179 (2009),1249-1268.
[7] M. Shabir, A. Khan and Y. Nawaz, Characterizations of regular semigroups by (╬▒, β)-fuzzy ideals, Comput. Math. Appl., 59 (2010), 161-175.
[8] M. Shabir and A. Khan, Fuzzy quasi-ideals of ordered semigroups, Bull. Malays. Math. Sci. Soc., 34(1) (2011), 87-102.
[9] M. Shabir and A. Khan, Characterizations of ordered semigroups by the properties of their fuzzy ideals, Comput. Math. Appl., 59(1) (2010), 539- 549.
[10] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems, 5 (1981), 203-215.
[11] N. Kuroki, On fuzzy semigroups, Inform. Sci., 53 (1991) 203-236.
[12] N. Kuroki, Fuzzy generalized bi-ideals in semigroups, Inform. Sci., 66 (1992), 235-243.
[13] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
[14] J. N. Mordeson, D. S. Malik and N. Kuroki, Fuzzy Semigroups, Studies in Fuzziness and Soft Computing, vol. 131, Springer-Verlag, Berlin, 2003.
[15] N.Kehayopulu, On intra-regular ordered semigroups, Semigroup Forum, 46 (1993), 271-278.
[16] N.Kehayopulu, On regular ordered semigroups, Math. Japonica, 45(3) (1997), 549-553.
[17] N.Kehayopulu and M.Tsingelis, On weakly prime ideals of ordered semigroups, Math. Japonica, 35(6) (1990), 1051-1056.
[18] N. Kehayopulu and M.Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum, 65 (2002), 128-132.
[19] N. Kehayopulu and M.Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci., 152 (2003), 231- 236.
[20] N. Kehayopulu and M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci., 171 (2005), 13-28.
[21] N. Kehayopulu and M. Tsingelis, Left regular and intra-regular ordered semigroups in terms of fuzzy subsets, Quasigroups and Related Systems, 14 (2006), 167-178.
[22] N. Kehayopulu and M. Tsingelis, Regular ordered semigroups in terms of fuzzy subsets, Inform. Sci., 176 (2006), 3675-3693.
[23] A. Khan, M. Shabir and Y. B. Jun, Intuitionistic fuzzy quasi-ideals of ordered semigroups, Russian Mathematics (Iz. VUZ), 54(12) (2010), 59- 71.
[24] J.von Neumann, On regular rings, Proc. Nat. Acad. Sci., 22 (1936), 707-713.
[25] L.Kov'acs, A note on regular rings, Publ. Math. Debrecen, 4 (1956), 465-468.
[26] K. Is'eki, A characterization of regular semigroups, Proc. Japan Acad., 32 (1956), 676-677.
[27] J.Calais, Demi-grupes quasi-inversifs, C. R. Acad. Sci. Paris, 252 (1961), 2357-2359.
[28] X. Y. Xie and J. Tang, Fuzzy radicals and prime fuzzy ideals of ordered semigroups, Inform. Sci., 178 (2008), 4357-4374.
[29] X. Y. Xie and J. Tang, Regular ordered semigroups and intra-regular ordered semigroups in terms of fuzzy subsets, Iranian Journal of Fuzzy Systems, 7(2) (2010), 121-140.
[30] X. Y. Xie, J. Tang and F. Yan, A characterization of prime fuzzy ideals of ordered semigroups, Fuzzy Systems and Mathematics, 22 (2008), 39-44.
[31] J. Tang, On completely semiprime, semiprime and prime fuzzy ideals in ordered semigroups, International Journal of Engineering and Natural Sciences, 5(1) (2011), 35-40.
[32] J. Tang, Chains of archimedean ordered semigroups by terms of fuzzy subsets, World Academy of Science, Engineering and Technology, 80 (2011), 957-963.
[33] X. Y. Xie, An introduction to ordered semigroup theory. Kexue Press, Beijing, 2001.
[34] X. Y. Xie and M. F. Wu, The Theory of Fuzzy Semigroups. Kexue Press, Beijing, 2005.