Main Article Content
Abstract
In this paper, we introduce the concept of generalized
fuzzy ideals in ternary semigroups, which is a generalization of the fuzzy
ideals of semigroups. In this regard, we define ($\alpha ,\beta $)-fuzzy
left (right, lateral) ideals, ($\alpha ,\beta $)-fuzzy quasi-ideals and ($
\alpha ,\beta $)-fuzzy bi-ideals and invetigate some related properties of
ternary semigroups. Special concentration is paid to ($\in ,\in \vee q$
)-fuzzy left (right, lateral) ideals, ($\in ,\in \vee q$)-fuzzy quasi-ideals
and ($\in ,\in \vee q$)-fuzzy bi-ideals. Finally, we
characterize regular ternary semigroups in terms of these notions.
Keywords
Article Details
References
- bibitem{1} J. Ahsan, Kui Yuan Li, M. Shabir, textit{Semigroups
- characterized by their fuzzy bi-ideals}, J. Fuzzy Math., 10(2) (2002) 441-449.
- bibitem{2} S. K. Bhakat, $left( in vee qright) $-textit{level subset},
- Fuzzy Sets Syst., 103 (1999) 529-533.
- bibitem{3} S. K. Bhakat, P. Das, textit{On the definition of a fuzzy
- subgroup}, Fuzzy Sets Syst., 51 (1992) 235-241.
- bibitem{4} S. K. Bhakat, P. Das, $left( in ,in vee qright) $-textit{%
- fuzzy subgroups}, Fuzzy Sets Syst., 80 (1996) 359-368.
- bibitem{5} S. K. Bhakat, P. Das, textit{Fuzzy subrings and ideals redefined}, Fuzzy Sets Syst., 81 (1996) 383-393.
- bibitem{6} P. S. Das, textit{Fuzzy groups and level subgroups}, J. Math.
- Anal. Appl., 84 (1981) 264-269.
- bibitem{7} B. Davvaz, $left( in ,in vee qright) $-textit{fuzzy
- subnearrings and ideals}, Soft Computing, 10 (2006) 206-211.
- bibitem{d} B. Davvaz, {it Fuzzy $R$-subgroups with thresholds of near-rings and implication operators}, Soft
- Computing, 12 (2008) 875-879.
- bibitem{8} V. N. Dixit, S. Dewan, textit{A note on quasi and
- bi-ideals in Ternary Semigroups}, Internat. J. Math. Math. Sci., 18(3) (1995) 501-508.
- bibitem{9} T. K. Dutta, S. Kar, textit{On ternary semifields,} Math.
- Gen. Algebra Appl., 24 (2004) 445-454.
- bibitem{10} T. K. Dutta, S. Kar, textit{On Prime ideals and Prime Radical
- of Ternary Semirings,} Bull. Cal. Math. Soc., 97(5)(2005) 185-198.
- bibitem{11} T. K. Dutta, S. Kar, textit{A note on regular ternary
- semirings}, Kyungpook Math. J., 46 (2006) 357-365.
- bibitem{12} A. Iampan, textit{The Minimality and Maximality of Left
- (Right) Ideals in Ternary Semigroups}, Int. J. Contem. Math. Sci., 5 (2010) 2409-2417.
- bibitem{13} Y. B. Jun, S. Z. Song, textit{Generalized fuzzy interior ideals
- in semigroups}, Information Sciences, 176 (2006) 3079-3093.
- bibitem{14} Y. B. Jun, textit{Generalization of }$(in ,in vee q)$textit{%
- -fuzzy sub algebras in BCK/BCI-algebras,} Comput. Math. Appl. 58 (2009)
- -1390.
- bibitem{15} Y. B. Jun, Y. Xu, J. Ma, textit{Redefined fuzzy implicative
- filters}, Information Sciences, 177 (2007) 1422-1429..
- bibitem{16} O. Kazanci, S. Yamak, textit{Generalized fuzzy bi-ideals of
- semigroup}, Soft Computing, 12 (2008) 1119-1124.
- bibitem{17} S.Kar, textit{On Ideals in ternary semirings}, Int.J. Math.
- Gen. Sci., 18 (2005) 3015-3023.
- bibitem{18} A. Khan, Y. B. Jun, M. Z. Abbas, textit{Characterizations of
- ordered semigroups in terms of (}$in ,in vee q$textit{)-fuzzy interior
- ideals}, Neural Computing and Applications, DOI 10.1007/s00521-010-0463-8.
- bibitem{19} N. Kuroki, textit{Fuzzy bi-ideals in semigroups}, Comment. Math.
- Univ. St. Pauli XXVIII-1 (1979) 17-21.
- bibitem{20} D. H. Lehmer, textit{A ternary analogue of abelian groups},
- Amer. J. Math., (1932) 329-338.
- bibitem{21} J. N. Mordeson, D. S. Malik and N. Kuroki, textit{Fuzzy
- Semigroups}, Studies in Fuzziness and Soft Computing, 131,
- Springer-Verlag Berlin (2003).
- bibitem{22} V. Murali, textit{Fuzzy points of equivalent fuzzy subsets,}
- Information Sciences, 158 (2004) 277-288.
- bibitem{24} M. Shabir, Y. B. Jun and Y. Nawaz,textit{ Semigroups characterized
- by }$(in ,in vee q)$textit{-fuzzy ideals,} Comput. Math. Appl., 60 (2010) 1473-1493.
- bibitem{25} M. Shabir, Y. B. Jun, Y. Nawaz, textit{Characterizations of
- regular semigroups by (}$alpha ,beta $textit{)-fuzzy ideals}, Comput.
- Math. Appli., 59 (2010) 161-175.
- bibitem{26} F. M. Sioson, textit{Ideal theory in ternary semigroups},
- Math.Japon., 10 (1965) 63-84.
- bibitem{27} L. A. Zadeh, textit{Fuzzy Sets}, Inform. & Control, 8 (1965)
- -353.
- bibitem{28} J. Zhan, B. Davvaz, K. P. Shum, textit{A new view of fuzzy
- hypernear-rings}, Information Sciences, 178 (2008) 425-438.
References
bibitem{1} J. Ahsan, Kui Yuan Li, M. Shabir, textit{Semigroups
characterized by their fuzzy bi-ideals}, J. Fuzzy Math., 10(2) (2002) 441-449.
bibitem{2} S. K. Bhakat, $left( in vee qright) $-textit{level subset},
Fuzzy Sets Syst., 103 (1999) 529-533.
bibitem{3} S. K. Bhakat, P. Das, textit{On the definition of a fuzzy
subgroup}, Fuzzy Sets Syst., 51 (1992) 235-241.
bibitem{4} S. K. Bhakat, P. Das, $left( in ,in vee qright) $-textit{%
fuzzy subgroups}, Fuzzy Sets Syst., 80 (1996) 359-368.
bibitem{5} S. K. Bhakat, P. Das, textit{Fuzzy subrings and ideals redefined}, Fuzzy Sets Syst., 81 (1996) 383-393.
bibitem{6} P. S. Das, textit{Fuzzy groups and level subgroups}, J. Math.
Anal. Appl., 84 (1981) 264-269.
bibitem{7} B. Davvaz, $left( in ,in vee qright) $-textit{fuzzy
subnearrings and ideals}, Soft Computing, 10 (2006) 206-211.
bibitem{d} B. Davvaz, {it Fuzzy $R$-subgroups with thresholds of near-rings and implication operators}, Soft
Computing, 12 (2008) 875-879.
bibitem{8} V. N. Dixit, S. Dewan, textit{A note on quasi and
bi-ideals in Ternary Semigroups}, Internat. J. Math. Math. Sci., 18(3) (1995) 501-508.
bibitem{9} T. K. Dutta, S. Kar, textit{On ternary semifields,} Math.
Gen. Algebra Appl., 24 (2004) 445-454.
bibitem{10} T. K. Dutta, S. Kar, textit{On Prime ideals and Prime Radical
of Ternary Semirings,} Bull. Cal. Math. Soc., 97(5)(2005) 185-198.
bibitem{11} T. K. Dutta, S. Kar, textit{A note on regular ternary
semirings}, Kyungpook Math. J., 46 (2006) 357-365.
bibitem{12} A. Iampan, textit{The Minimality and Maximality of Left
(Right) Ideals in Ternary Semigroups}, Int. J. Contem. Math. Sci., 5 (2010) 2409-2417.
bibitem{13} Y. B. Jun, S. Z. Song, textit{Generalized fuzzy interior ideals
in semigroups}, Information Sciences, 176 (2006) 3079-3093.
bibitem{14} Y. B. Jun, textit{Generalization of }$(in ,in vee q)$textit{%
-fuzzy sub algebras in BCK/BCI-algebras,} Comput. Math. Appl. 58 (2009)
-1390.
bibitem{15} Y. B. Jun, Y. Xu, J. Ma, textit{Redefined fuzzy implicative
filters}, Information Sciences, 177 (2007) 1422-1429..
bibitem{16} O. Kazanci, S. Yamak, textit{Generalized fuzzy bi-ideals of
semigroup}, Soft Computing, 12 (2008) 1119-1124.
bibitem{17} S.Kar, textit{On Ideals in ternary semirings}, Int.J. Math.
Gen. Sci., 18 (2005) 3015-3023.
bibitem{18} A. Khan, Y. B. Jun, M. Z. Abbas, textit{Characterizations of
ordered semigroups in terms of (}$in ,in vee q$textit{)-fuzzy interior
ideals}, Neural Computing and Applications, DOI 10.1007/s00521-010-0463-8.
bibitem{19} N. Kuroki, textit{Fuzzy bi-ideals in semigroups}, Comment. Math.
Univ. St. Pauli XXVIII-1 (1979) 17-21.
bibitem{20} D. H. Lehmer, textit{A ternary analogue of abelian groups},
Amer. J. Math., (1932) 329-338.
bibitem{21} J. N. Mordeson, D. S. Malik and N. Kuroki, textit{Fuzzy
Semigroups}, Studies in Fuzziness and Soft Computing, 131,
Springer-Verlag Berlin (2003).
bibitem{22} V. Murali, textit{Fuzzy points of equivalent fuzzy subsets,}
Information Sciences, 158 (2004) 277-288.
bibitem{24} M. Shabir, Y. B. Jun and Y. Nawaz,textit{ Semigroups characterized
by }$(in ,in vee q)$textit{-fuzzy ideals,} Comput. Math. Appl., 60 (2010) 1473-1493.
bibitem{25} M. Shabir, Y. B. Jun, Y. Nawaz, textit{Characterizations of
regular semigroups by (}$alpha ,beta $textit{)-fuzzy ideals}, Comput.
Math. Appli., 59 (2010) 161-175.
bibitem{26} F. M. Sioson, textit{Ideal theory in ternary semigroups},
Math.Japon., 10 (1965) 63-84.
bibitem{27} L. A. Zadeh, textit{Fuzzy Sets}, Inform. & Control, 8 (1965)
-353.
bibitem{28} J. Zhan, B. Davvaz, K. P. Shum, textit{A new view of fuzzy
hypernear-rings}, Information Sciences, 178 (2008) 425-438.