Abstract
In this paper, we introduce a considerable machinery that permits us to characterize a number of special (fuzzy) subsets in ordered semigroups. In this regard, we generalize (Davvaz and Khan in Inform Sci 181:1759–1770 2011) and define (\(\in ,\in \vee q_{k}\))-fuzzy generalized bi-ideals in ordered semigroups, which is a generalization of the concept of an (α, β)-fuzzy generalized bi-ideal in an ordered semigroup. We also define (\(\in ,\in \vee q_{k}\))-fuzzy left (resp. right)-ideals. Using these concept, some characterization theorems of regular, left (resp. right) regular, completely regular and weakly regular ordered semigroups are provided. The upper/lower parts of an (\(\in ,\in \vee q_{k}\))-fuzzy generalized bi-ideal and (\(\in ,\in \vee q_{k}\))-fuzzy left (resp. right)-ideal are given, and some characterizations are provided.
Similar content being viewed by others
References
Bhakat SK, Das P (1996) \(\left( \in ,\in \vee q\right) \)-fuzzy subgroups. Fuzzy Sets Syst 80:359–368
Bhakat SK, Das P (1996) Fuzzy subrings and ideals redefined. Fuzzy Sets Syst 81:383–393
Davvaz B, Khan A (2011) Characterizations of regular ordered semigroups in terms of \(\left( \alpha,\beta \right) \)-fuzzy generalized bi-ideals. Inform Sci 181:1759–1770
Davvaz B, Mozafar Z (2009) (\(\in ,\in \vee \)q)-fuzzy Lie subalgebra and ideals. Int J Fuzzy Syst 11(2):123–129
Davvaz B (2008) Fuzzy R-subgroups with thresholds of near-rings and implication operators. Soft Comput 12:875–879
Davvaz B (2006) (\(\in ,\in \vee \)q)-fuzzy subnear-rings and ideals. Soft Comput 10:206–211
Jun YB, Khan A, Shabir M (2009) Ordered semigroups characterized by their (\(\in ,\in \vee \)q)-fuzzy bi-ideals. Bull Malays Math Sci Soc 32(3):391–408
Jun YB, Song SZ (2006) Generalized fuzzy interior ideals in semigroups. Inform Sci 176:3079–3093
Jun YB (2005) On (α, β)-fuzzy subalgebras of BCK/BCI-algebras. Bull Korean Math Soc 42(4):703–711
Jun YB (2007) Fuzzy subalgebras of type (α, β) in BCK/BCI-algebras. Kyungpook Math J 47(3):403–410
Jun YB, Dudek WA, Shabir M, Kang MS (2010) General form of (α, β)-fuzzy ideals of Hemirings. Honam Math J 32(3):413–439
Khan A, Jun YB, Mahmood T (2010) Generalized fuzzy interior ideals of Abel Grassmann’s groupoids. Int J Math Math Sci. Article ID 838392, 14
Kazanci O, Yamak S (2008) Generalized fuzzy bi-ideals of semigroup. Soft Comput 12:1119–1124
Kehayopulu N, Tsingelis M (2005) Fuzzy bi-ideals in ordered semigroups. Inf Sci 171:13–28
Khan A, Jun YB, Abbas Z Characterizations of ordered semigroups in terms of (\(\in ,\in \vee q\))-fuzzy interior ideals. Neural Comput Appl. doi:10.1007/s00521-010-0463-8
Kuroki N (1981) On fuzzy ideals and fuzzy bi-ideals in semigroups. Fuzzy Sets Syst 5:203–215
Kuroki N (1993) Fuzzy semiprime quasi-ideals in semigroups. Inf Sci 75:201–211
Kuroki N (1991) On fuzzy semigroups. Inf Sci 53:203–236
Lee SK, Park KY (2003) On right (left) duo po-semigroups. Kangweon-Kyungki Math J 11(2):147–153
Mordeson JN, Malik DS and Kuroki N (2003) Fuzzy semigroups. Studies in fuzziness and soft computing, vol 131. Springer, Berlin
Murali V (2004) Fuzzy points of equivalent fuzzy subsets. Inf Sci 158:277–288
Pu PM, Liu YM (1980) Fuzzy topology I, neighborhood structure of a fuzzy point and Moore-Smith convergence. J Math Anal Appl 76:571–599
Rosenfeld A (1971) Fuzzy groups. J Math Anal Appl 35:512–517
Shabir M, Khan A (2008) Characterizations of ordered semigroups by the properties of their fuzzy generalized bi-ideals. New Math Nat Comput 4(2):237–250
Shabir M, Jun YB, Nawaz Y (2010) Semigroups characterized by (α, β)-fuzzy ideals. Comput Math Appl 59:161–175
Shabir M, Jun YB, Nawaz Y (2010) Characterization of regular semigroups by (\(\in ,\in \vee q_{k}\))-fuzzy ideals. Comput Math Appl 60:1473–1493
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khan, A., Jun, Y.B., Sarmin, N.H. et al. Ordered semigroups characterized by (\({ \in,\in \vee q}_{k}\))-fuzzy generalized bi-ideals. Neural Comput & Applic 21 (Suppl 1), 121–132 (2012). https://doi.org/10.1007/s00521-011-0731-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00521-011-0731-2