Generalized Roughness of (∈, ∈ ∨q)-Fuzzy Ideals in Ordered Semigroups

Yıl 2019, Sayı: 26, 32 - 53, 01.01.2019

Azmat Hussain Muhammad İrfan Ali Tahir Mahmood

Öz

Ordered semigroups (OSGs) is a significant algebraic structure having partial ordered with associative binary operation. OSGs have broad applications in various fields such as coding theory, automata theory, fuzzy finite state machines and computer science etc. In this manuscript we investigate the notion of generalized roughness for fuzzy ideals in OSGs on the basis of isotone and monotone mappings. Then the notion of approximation is boosted to the approximation of fuzzy bi-ideals,~approximations fuzzy interior ideals and approximations fuzzy quasi-ideals in OSGs and investigate their related properties. Furthermore  (\isin;,\isin;\or;q)-fuzzy ideals are the generalization of fuzzy ideals. Also the generalized roughness for (\isin;,\isin;\or;q)-fuzzy ideals, fuzzy bi-ideals and fuzzy interior ideals have been studied in OSGs and discuss the basic properties on the basis of isotone and monotone mappings

Anahtar Kelimeler

Fuzzy sets, Rough sets, Approximations of fuzzy ideals

Kaynakça

• [1] J. Ahsan, R. M. Latif, M. Shabir, Fuzzy quasi-ideals in semigroup, Journal of Fuzzy Mathematics 9 (2001) 259-270.
• [2] M. I. Ali, T. Mahmood, A. Hussain, A study of generalized roughness in -fuzzy flters of ordered semigroups, Journal of Taibah University for Science 12 (2018) 163-172.
• [3] S. K. Bhakat, P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems 51 (1992) 235-241.
• [4] S. K. Bhakat, P. Das, (2;2 _q) fuzzy subgroups, Fuzzy Sets and Systems 80 (1996) 359-368.
• [5] S. K. Bhakat, P. Das, Fuzzy subrings and ideals refined, Fuzzy Sets and Systems 81 (1996) 383-393.
• [6] B. Davvaz, A short note on algebra T-rough sets, Information Sciences 178 (2008) 3247-3252:
• [7] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System 17 (1990) 191-208.
• [8] Y. B. Jun, S. Z. Song, Generalized fuzzy interior ideal in semigroup, Information Science 176 (2006) 3079-3093:
• [9] Y. B. Jun, A. Khan, M. Shabir, Ordered semigroups characterized by their (2;2q) fuzzy bi-ideals, Bulletin of the Malaysian Mathematical Sciences Society 32 (2009) 391-408.
• [10] Y. B. Jun, W. A. Dudek, M. Shabir, M. S. Kang, General form of fuzzy ideals of Hemirings, Honam Mathematical Journal 32 (2010) 413-439.
• [11] O. Kazanci, B. Davvaz, On the structure of rough prime ideals and rough fuzzy prime ideals in commutative ring, Information Sciences 178 (2008) 1343-1354.
• [12] A. Khan, M. Shabir, (®; ¯) fuzzy interior ideals in ordered semigroups, Lobachevskii Journal of Mathematics 30 (2009) 30-39.
• [13] N. Kehayopulu, Remark on ordered semigroups, Math Japonica, 35 (1990) 1061-1063.
• [14] N. Kehayopulu, M. Tsingelis, A note on fuzzy sets in semigroups, Scientiae Mathematicae 2 (1999) 411-413.
• [15] N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128-132.
• [16] N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Information Science 171 (2005) 13-28.
• [17] N. Kehayopulu, M. Tsingelis, Fuzzy right, left, quasi-ideals, bi-ideals in ordered semigroups, Lobachevskii Journal of Mathematics 30 (2009) 17-22.
• [18] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5 (1981) 203-215.
• [19] N. Kuroki, On fuzzy semigroups, Information Science 53 (1991) 203-236.
• [20] N. Kuroki, Fuzzy semiprime quasi-ideals in semigroups, Information Science 75 (1993) 201-211.
• [21] N. Kuroki, Rough Ideals in Semigroups, Information Sciences 100 (1997) 139- 163.
• [22] T. Mahmood, M. I. Ali, and A. Hussain, Generalized roughness in fuzzy ¯l- ters and fuzzy ideals with thresholds in ordered semigroups, Computational and Applied Mathematics (2018) 1-21.
• [23] Z. Pawlak, Rough sets, International Journal of Computers Science 11 (1982) 341-356.
• [24] P. Pu, Y. Liu, Fuzzy topology l. neighborhood structure of a fuzzy point and Moore-Smith convergence, Journal of Mathematical Analysis and Applications 76 (1980) 571-599.
• [25] S. M. Qurashi, and M. Shabir, Generalized approximations of (in; 2q)-fuzzy ideals in quantales, Computational and Applied Mathematics (2018) 1-17.
• [26] N. Rehman, N. Shah, M. I. Ali, A. Ali, Generalised roughness in (2;2q)-fuzzy substructures of LA-semigroups, Journal of the National Science Foundation of Sri Lanka 3 (2018).
• [27] A. Rosenfeld, Fuzzy groups, Journal of Mathematical Analysis and Applications 35 (1971) 512-517.
• [28] M. Shabir, T. Mahmood, Spectrum of (2;2 _q)-fuzzy prime h-ideals of a hemiring, World Applied Science Journal 17 (2012) 1815-1820.
• [29] M. Shabir, T. Mahmood S. Hussain, Hemirings characterized by interval valued (2;2 _q)-fuzzy k-ideals, World Applied Sciences Journal 20 (2012) 1678-1684.
• [30] M. Shabir, Y. Nawaz, T. Mahmood, Characterizations of hemirings by (2;2 _q)-fuzzy ideals, East Asian Mathematical Journal 31 (2015) 001-018.
• [31] Q. M. Xiao, Z. L. Zhang, Rough prime ideals and rough fuzzy prime ideals in semigroups, Information Sciences 176 (2006) 725-733.
• [32] N. Yaqoob, S. Abdullah, N. Rehman, M. Naeem, Roughness and fuzziness in ordered ternary semigroups, World Applied Sciences Journal 12 (2012) 1683-1693.
• [33] N. Yaqoob, M. Aslam, Generalized rough approximations in ¡-semihypergroups, Journal of Intelligent and Fuzzy Systems 27 (2014) 2445-2452.
• [34] N. Yaqoob, M. Aslam, K. Hila, B. Davvaz, Rough prime bi-¡-hyperideals and fuzzy prime bi-¡-hyperideals of ¡-semihypergroups, Filomat 31 (2017) 4167-4183.
• [35] N. Yaqooob, I. Rehman, M. Aslam, Approximations of bipolar fuzzy ¡-hyperideals of ¡-semihypergroups, Afrika Matematika 29 (2018) 869-886.
• [36] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353.
• [37] J. Zhan, N. Yaqoob, M. Khan, Roughness in non-associative po-semihyprgroups based on pseudohyperorder relations, Journal of Multiple-Valued Logic and Soft Computing 28 (2017) 153-177.