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Ideals and fuzzy ideals on residuated lattices

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Abstract

This paper mainly focus on building the ideals theory of non regular residuated lattices. Firstly, the notions of ideals and fuzzy ideals of a residuated lattice are introduced, their properties and equivalent characterizations are obtained; at the meantime, the relation between filter and ideal is discussed. Secondly, two types prime ideals of a residuated lattice are introduced, the relations between the two types ideals are studied, in some special residuated lattices (such as MTL-algebras, lattice implication algebras, BL-algebras), prime ideal and prime ideal of the second kind are coincide. At the meantime, the notions of fuzzy prime ideal and fuzzy prime ideal of the second kind on a residuated lattice are introduced, aiming at the relation between prime ideal and prime ideal of the second kind, we mainly investigate the fuzzy prime ideal of the second kind. Finally, we investigated the fuzzy congruence relations induced by fuzzy ideal, we construct a new residuated lattice induced by fuzzy congruences, the homomorphism theorem is given.

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References

  1. Akram M, Davvaz B (2012) Generalized fuzzy ideals of K-algebras. J Multi Value Logic Soft Comput 19:475–491

    MathSciNet  Google Scholar 

  2. Borzooei RA, Khosravani Shoar S, Ameri R (2012) Some types of filters in MTL-algebras. Fuzzy Sets Syst 187:92–102

    Article  MathSciNet  MATH  Google Scholar 

  3. Belohlavek R (2003) Some properties of residuated lattices. Czechoslovak Math J 53:161–171

    Article  MathSciNet  MATH  Google Scholar 

  4. Dilworth RP, Ward M (1939) Residuated lattices. Trans Am Math Soc 45:335–354

    Article  MathSciNet  MATH  Google Scholar 

  5. Farahani Hadi, Zahiri Omid (2013) Algebraic view of MTL-filters. Ann Univ Craiova Math Comput Sci Ser 40(1):34–44

    MathSciNet  MATH  Google Scholar 

  6. Van Gasse B, Deschrijver G, Cornelis C, Kerre EE (2010) Filters of residuated lattices and triangle algebras. Inf Sci 180:3006–3020

    Article  MathSciNet  MATH  Google Scholar 

  7. Hájek P (1998) Metamathematics of fuzzy logic. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  8. Hedayati H (2013) A generalization of (implicative) (\(\in\), \(\in \vee q\))-fuzzy ideals of pseudo-MV algebras. J Multi Value Logic Soft Comput 20:625–651

    MathSciNet  Google Scholar 

  9. Jun YB, Xu Y, Zhang XH (2005) Fuzzy filters of MTL-algebras. Inf Sci 175:120–138

    Article  MathSciNet  MATH  Google Scholar 

  10. Kondo M, Dudek WA (2008) Filter theory of BL-algebras. Soft Comput 12:419–423

    Article  MATH  Google Scholar 

  11. Kondo M (2010) Filters on commutative residuated lattices. Adv Intell Soft Comput 68:343–347

    MATH  Google Scholar 

  12. Kondo M, Turunen E (2012) Prime filters on residuated lattices. In: 2012 IEEE 42nd international symposium on multiple-valued logic, pp 89–91

  13. Lele C, Nganou JB (2013) MV-algebras derived from Ideals in BL-algebras. Fuzzy Sets Syst 218:103–113

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu Y, Xu Y, Qin XY (2014) Interval-valued \(\cal {T}\) -fuzzy filters and interval-valued \(\cal {T}\) -fuzzy congruences on residuated lattices. J Intell Fuzzy Syst 26:2021–2033

    MathSciNet  MATH  Google Scholar 

  15. Liu Y, Xu Y, Jia HR (2013) Determination of 3-ary-resolution in lattice-valued propositional logic LP(X). Int J Comput Intell Syst 6:943–953

    Article  Google Scholar 

  16. Liu Y, Xu Y, Qin X (2014) Interval-valued intuitionistic (TS)-fuzzy filters theory on residuated lattices. Int J Mach Learn Cybern 5(5):683–696

    Article  Google Scholar 

  17. Pei DW (2002) The characterization of residuated lattices and regular residuated lattices. Acta Math Sinica 42(2):271–278

    MathSciNet  MATH  Google Scholar 

  18. Wang GJ (2000) Non-classical mathenatical logic and approximating reasoning. Science Press, Beijing

    Google Scholar 

  19. Xu Y, Ruan D, Qin K, Liu J (2003) Lattice-valued logic. Springer, Berlin

    Book  MATH  Google Scholar 

  20. Zhang XH, Li WH (2005) On fuzzy logic algebraic system MTL. Adv Syst Sci Appl 5:475–483

    Google Scholar 

  21. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhan JM, Xu Y (2008) Some types of generalized fuzzy filters of BL-algebras. Comput Math Appl 56:1064–1616

    MathSciNet  MATH  Google Scholar 

  23. Zhu H, Zhao JB, Xu Y (2013) IFI-ideals of lattice implication algebras. Int J Comput Intell Syst 6:1002–1011

    Article  Google Scholar 

  24. Zhu YQ, Liu YL, Xu Y (2010) On \((\in ,\in \vee q)\) -fuzzy filters of residuated lattices. In: Ruan D et al (eds) Computational intelligence foundations and applications, vol 4. World Scientific Publishing Co., Pte. Ltd, Singapore, pp 79–84

  25. Zhu YQ, Xu Y (2010) On filter theory of residuated lattices. Inf Sci 180:3614–3632

    Article  MathSciNet  MATH  Google Scholar 

  26. Xue ZA, Xiao YH, Liu WH, Cheng HR, Li YJ (2013) Intuitionistic fuzzy filter theory of BL-algebras. Int J Mach Learn Cybern 4(6):659–669

    Article  Google Scholar 

  27. Zou L, Liu X, Pei Z, Huang D (2013) Implication operators on the set of \(\vee\)-irreducible element in the linguistic truth-valued intuitionistic fuzzy lattice. Int J Mach Learn Cybern 4(4):365–372

    Article  Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous reviewers’ comments and suggestions improved both content and the presentation of this paper. One of reviewer point out many typing mistakes and grammar mistakes in the manuscripts, we gave him (her) heartfelt thanks. This work was supported by National Natural Science Foundation of P.R.China (Grant No. 61175055, 61305074), Sichuan Key Technology Research and Development Program (Grant No.2011FZ0051), Sichuan Province Science and technology plan project (No. 15JC0239), Radio Administration Bureau of MIIT of China (Grant No. [2011]146), China Institution of Communications (Grant No. [2011]051). The Speciality Comprehensive Reform of Mathematics and Applied Mathematics of Ministry of Education (ZG0464). The Speciality Comprehensive Reform of Mathematics and Applied Mathematics of Ministry of Education (01249). A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department (14ZA0245). The Scientific Research Research Fund of Neijiang Normal University (No. 13ZB05).

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Authors and Affiliations

  1. College of Mathematics and Information Sciences, Neijiang Normal University, Neijiang, Sichuan, 641000, China

    Yi Liu & Ya Qin

  2. College of Mathematics and Computer Sciences, Shanxi Norm University, Linfen, Shanxi, 041000, China

    Xiaoyan Qin

  3. Intelligent Control Development Center, Southwest Jiaotong University, Chengdu, Sichuan, 610031, China

    Yang Xu

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  1. Yi Liu
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  2. Ya Qin
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  3. Xiaoyan Qin
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  4. Yang Xu
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Correspondence to Yi Liu.

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Liu, Y., Qin, Y., Qin, X. et al. Ideals and fuzzy ideals on residuated lattices. Int. J. Mach. Learn. & Cyber. 8, 239–253 (2017). https://doi.org/10.1007/s13042-014-0317-2

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  • DOI: https://doi.org/10.1007/s13042-014-0317-2

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