Skip to main content
SpringerLink
Log in

On two new classes of stabilizers in residuated lattices

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this paper, we introduce some stabilizers and study related properties of them in residuated lattices. Then, we investigate the image and inverse image of a right and left stabilizer of a nonempty subset under a homomorphism. Besides, we discuss the relations between stabilizers and several special filters (ideals) in residuated lattices. Moreover, we also characterize some special classes of residuated lattices, for example, Heyting algebras and linearly ordered Heyting algebras, in terms of these stabilizers. Finally, we discuss the relations between these stabilizers and get that the right implicative stabilizers and right multiplicative stabilizers are order isomorphic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Balbes R, Dwinger P (1974) Distributive lattices. University of Missouri Press, Columbia

    MATH  Google Scholar 

  • Blount K, Tsinakis C (2003) The structure of residuated lattices. Int J Algebra Comput 13:437–461

    Article  MathSciNet  Google Scholar 

  • Borzooei RA, Paad A (2012) Some new types of stabilizers in \(BL\)-algebras and their applications. Indian J Sci Technol 5:1910–1915

    Google Scholar 

  • Chen X, Xin X, Wang J (2017) Fuzzy stabilizers in MV-algebras. In: 13th international conference on natural computation, fuzzy systems and knowledge discovery (ICNC-FSKD). IEEE, Guilin, China, pp 1122–1126

  • Chen X, Xin X, Wang J (2018) Fuzzy stabilizers in \(BL\)-algebras. Filomat 32(5):1783–1788

    Article  MathSciNet  Google Scholar 

  • Deng W, Zhao H, Zou L, Li G, Yang X, Wu D (2017a) A novel collaborative optimization algorithm in solving complex optimization problems. Soft Comput 21:4387–4398

    Article  Google Scholar 

  • Deng W, Zhao H, Yang X, Xiong J, Sun M, Li B (2017b) Study on an improved adaptive PSO algorithm for solving multi-objective gate assignment. Appl Soft Comput 59:288–302

    Article  Google Scholar 

  • Deng W, Xu J, Zhao H (2019a) An improved ant colony optimization algorithm based on hybrid strategies for scheduling problems. IEEE Access 7:20281–20292

    Article  Google Scholar 

  • Deng W, Yao R, Zhao H, Yang X, Li G (2019b) A novel intelligent diagnosis method using optimal LS-SVM with improved PSO algorithm. Soft Comput 23:2445–2462

    Article  Google Scholar 

  • Dilworth RP (1939) Non-commutative residuated lattices. Trans Am Math Soc 46:426–444

    Article  MathSciNet  Google Scholar 

  • Forouzesh F, Eslami E, Borumand A (2014) Saeid, stabilizer theory in \(MV\)-algebras. J Intell Fuzzy Syst 27:1557–1565

    MATH  Google Scholar 

  • Grätzer G (1979) Lattice theory. W.H. Freeman and Company, San Francisco

    MATH  Google Scholar 

  • Hájek G (1998) Metamathematics of fuzzy logic, trends in logic-studia logica library, vol 4. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Haveshki M, Mohamadhasani M (2010) Stabilizer in \(BL\)-algebras and its properties. Int Math 5(57–60):2809–2816

    MathSciNet  MATH  Google Scholar 

  • Höhle U (1995) Commutative, residuated \(l\)-monoids. In: Höhle U, Klement E (eds) Non-classical logics and their applications to fuzzy subsets. Kluwer, Dordrecht, pp 53–106

    Chapter  Google Scholar 

  • Idziak PM (1984) Lattice operations in \(BCK\)-algebras. Math Jpn 29:839–846

    MathSciNet  MATH  Google Scholar 

  • Kowalski T, Ono H (2001) Residuated lattices: an algebraic glimpse at logic without contraction. Elsevier

  • Krull W (1924) Axiomatische Begründung der allgemeinen Ideal theorie. Sitzungsber Physik Med Soc Erlangen 56:47–63

    MATH  Google Scholar 

  • Liu LZ, Li KT (2007) Boolean filters and positive implicative filters of residuated lattices. Inf Sci 177:5725–5728

    Article  MathSciNet  Google Scholar 

  • Motamed S (2017) Double left stabilizers in \(BL\)-algebras. Ann Univ Crai Math Comput Sci Ser 44(2):214–227

    MathSciNet  MATH  Google Scholar 

  • Motamed S, Torkzadeh L (2017) A new class of \(BL\)-algebras. Soft Comput 21:687–698

    Article  Google Scholar 

  • Oksda M, Terui K (1999) The finite model property for various fragments of intuitionistic linear login. J Symb Log 64:790–802

    Article  Google Scholar 

  • Pavelka J (1979) On fuzzy logic II. Enriched residuated lattices and semantics of propositional calculi. Z Math Log Grundl Math 25:119–134

    Article  MathSciNet  Google Scholar 

  • Piciu D (2007) Algebras of fuzzy logic, Ed. Universitaria, Craiova

  • Roudabri T, Torkzadeh L (2009) A topology on \(BCK\)-algebras via left and right stabilizers. Iran J Math Sci Inf 4:1–8

    MathSciNet  MATH  Google Scholar 

  • Turumen E (1999) Mathematics behind fuzzy logic. Physica-Verlag, Heidelberg

    Google Scholar 

  • Wang JT, He PF, Saeid AB (2018) Stabilizers in MTL-algebras. J Intell Fuzzy Syst 35:717–727

    Article  Google Scholar 

  • Wang JT, Xin X, Saeid AB (2016) Very true on MTL-algebras. Open Math 14:955–969

    MathSciNet  MATH  Google Scholar 

  • Ward M (1940) Residuatd distributive lattices. Duke Math J 6:641–651

    Article  MathSciNet  Google Scholar 

  • Ward M, Dilworth RM (1939) Residuatd lattice. Trans Am Math Soc 45:335–354

    Article  Google Scholar 

  • Xie M (2014) Fuzzy stabilizer in \(IMTL\)-algebras. Appl Math Inf Sci 8(5):2479–2484

    Article  MathSciNet  Google Scholar 

  • Zahiri S, Saeid AB, Eslami E (2018) A study stabilizers in triangle algebras. Math Slovaca 68(1):41–52

    Article  MathSciNet  Google Scholar 

  • Zhao H, Sun M, Deng W, Yang X (2017) A new feature extraction method based on EEMD and multi-scale fuzzy entropy for motor bearing. Entropy. https://doi.org/10.3390/e19010014

    Article  Google Scholar 

  • Zhao H, Yao R, Xu L, Yuan Y, Li G, Deng W (2018) Study on a novel fault damage degree identification methods using high-order differential mathematical morphology gradient spectrum entropy. Entropy. https://doi.org/10.3390/e20090682

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere thanks to the editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. This work was supported in part by Higher Education Key Scientific Research Program Funded by Henan Province (Nos. 18A110008, 18A110010, 18A630001) and Research and Cultivation Fund Project of Anyang Normal University (Nos. AYNUKP-2018-B25, AYNUKP-2018-B26).

Author information

Authors and Affiliations

  1. School of Information and Mathematics, Yangtze University, Jingzhou, People’s Republic of China

    Kuanyun Zhu

  2. School of Business Administration, Hunan University, Changsha, People’s Republic of China

    Jingru Wang

  3. School of Mathematics and Statistics, Anyang Normal University, Anyang, People’s Republic of China

    Yongwei Yang

Authors
  1. Kuanyun Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jingru Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongwei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jingru Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed any of the authors.

Additional information

Communicated by A. Di Nola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, K., Wang, J. & Yang, Y. On two new classes of stabilizers in residuated lattices. Soft Comput 23, 12209–12219 (2019). https://doi.org/10.1007/s00500-019-04204-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-04204-y

Keywords

Search

Navigation