Abstract
The central objective of this paper is to introduce \(\left( {\alpha ,\beta } \right)\)-bipolar fuzzy ideals (left, lateral and right) and bi-ideals in ternary semirings by applying the definitions of belongingness \(\left( \in \right)\) and quasi-coincidence (\(q\)) of a bipolar fuzzy point with a bipolar fuzzy set. In this work, upper parts and lower parts of bipolar fuzzy set in ternary semirings are also discussed. Regular and intra-regular ternary semirings in terms of \(\left( { \in, \in \vee q} \right)\)-bipolar fuzzy (left, lateral and right) ideals and \(\left( { \in, \in \vee q} \right)\)-bipolar fuzzy bi-ideals are characterized.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akram M, Dar KH, Shum KP (2011) Interval-valued (α, β)-fuzzy K-algebras. Appl Soft Comput 11(1):1213–1222
Bashir S, Fatima M, Shabir M (2019) Regular ordered ternary semigroups in terms of bipolar fuzzy ideals. Mathematics 7(233):1–16
Bhakat SK, Das P (1992) On the definition of a fuzzy subgroup. Fuzzy Sets Syst 51(2):235–241
Daddi VR (2017) Intra-regular ternary semigroups. Glob J Pure Appl Math 13(10):7689–7694
Dutta TK, Kar S (2003) On regular ternary semirings, advances in algebra. In: Proceedings of the ICM satellite conference in algebra and related topics, World Scientific (2003), pp 343–355
Dutta TK, Kar S (2006) A note on regular ternary semirings. Kyungpook Math J 46(3):357
Golan JS (1999) Semirings and their applications. Kluwer Academic Publications, Dordrecht
Hayat K, Liu XC, Cao BY (2016) Bipolar fuzzy BRK-ideals in BRK-algebras. In: International workshop on mathematics and decision science, vol 646. Springer, Cham, pp 3–15
Hayat K, Mahmood T, Cao BY (2017) On bipolar anti fuzzy h-ideals in hemi-rings. Fuzzy Inf Eng 9:1–19
Hebisch U, Weinert HJ (1998) Semirings: algebraic theory and applications in computer science, vol 5. World Scientific, Singapore
Ibrar M, Khan A, Davvaz B (2017) Characterizations of regular ordered semigroups in terms of (α, β)-bipolar fuzzy generalized bi-ideals. J Intell Fuzzy Syst 33(1):365–376
Jan N, Zedam L, Mahmood T, Ullah K (2019) Cubic bipolar fuzzy graphs with applications. J Intell Fuzzy Syst 37(2):2289–2307
Jun YB, Kavikumar J (2011) Bipolar fuzzy finite state machines. Bull Malays Math Sci Soc 34(1):181–188
Jun YB, Kang MS, Kim HS (2009) Bipolar fuzzy structures of some types of ideals in hyper BCK-algebras. Sci Math Jpn 70:109–121
Kar S (2005) On quasi-ideals and bi-ideals in ternary semirings. Int J Math Math Sci 18:3015–3023
Kavikumar J, Khamis AB (2007) Fuzzy ideals and fuzzy quasi-ideals in ternary semirings. Int J Appl Math 37(2):98
Kavikumar J, Khamis A, Jun YB (2009) Fuzzy bi-ideals in ternary semirings. Int J Math Stat Sci 1:54–58
Lee KM (2000) Bipolar-valued fuzzy sets and their operations. In: Proceedings of the international conference on intelligent technologies, Bangkok, Thailand, pp 307–312
Lehmer DH (1932) A ternary analogue of abelian groups. Am J Math 54(2):329–338
Mahmood T, Ejaz A (2015) On bipolar valued fuzzy k-ideals in hemirings. Nucleus 52(3):115–122
Mahmood T, Hayat K (2015) Characterizations of hemi-rings by their bipolar-valued fuzzy h-ideals. Inf Sci Lett 4(2):51–59
Mahmood T, Munir M (2013) On bipolar fuzzy subgroups. World Appl Sci J 27(12):1806–1811
Mordeson JN, Malik DS (1998) Fuzzy commutative algebra. World Scientific, Springer
Nandi AK (2012) GA-fuzzy approaches: application to modeling of manufacturing process. In Statistical and computational techniques in manufacturing. Springer, Berlin, pp 145–185
Rao DM, Rao GS (2014) Special elements of a ternary semiring. Int J Eng Res Appl 4(11):123–130
Rosenfeld A (1971) Fuzzy groups. J Math Anal Appl 35(3):512–517
Shabir M, Liaqat S, Bashir S (2019) Regular and intra-regular semirings in terms of bipolar fuzzy ideals. Comput Appl Math 38:197
Ullah K, Mahmood T, Jan N, Broumi S, Khan Q (2018) On bipolar-valued hesitant fuzzy sets and their applications in multi-attribute decision making. Nucleus 55(2):93–101
Yang XP, Hayat K, Wang PH, Zhou XG (2016) Note on Max-Łukasiewicz bipolar fuzzy relation equation. In: International workshop on mathematics and decision science. Springer, Cham, pp 210–219
Ying-Ming L, Mao-Kang L (1998) Fuzzy topology, vol 9. World Scientific, Singapore
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Zhan J, Dudek WA (2007) Fuzzy h-ideals of hemirings. Inf Sci 177(3):876–886
Zhang WR (1994) Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Fuzzy information processing society biannual conference, 1994. Industrial fuzzy control and intelligent systems conference, and the NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, (1994), pp 305–309
Zhou M, Li S (2014) Application of bipolar fuzzy sets in semirings. J Math Res Appl 34(1):61–72
Acknowledgements
The authors would like to thank the Senior Editor, Subbulakshmi, Associate Editor, Anibal Tavares de Azevedo and the reviewers who have generously given their valuable time to review and improve this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anibal Tavares de Azevedo.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bashir, S., Mazhar, R., Abbas, H. et al. Regular ternary semirings in terms of bipolar fuzzy ideals. Comp. Appl. Math. 39, 319 (2020). https://doi.org/10.1007/s40314-020-01319-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01319-z
Keywords
- Ternary semiring
- (α, β)-Bipolar fuzzy subsemirings
- (α, β)-Bipolar fuzzy ideals
- Regular and intra-regular ternary semiring