Soft Computing

, Volume 23, Issue 10, pp 3207–3216 | Cite as

\(\alpha \)-filters and prime \(\alpha \)-filter spaces in residuated lattices

  • Yan Yan Dong
  • Xiao Long XinEmail author


The main goal of this paper is to introduce and research co-annihilators and \(\alpha \)-filters in residuated lattices. We characterize \(\alpha \)-filters in terms of co-annihilators. Also, we answer the open problem which appeared in Haveshki and Mohamadhasani (J Intell Fuzzy Syst 28:373–382, 2015). We prove that the lattice of all \(\alpha \)-filters in a residuated lattice forms a complete Heyting algebra. Finally, we investigate some topological properties of space of prime \(\alpha \)-filters and give necessary and sufficient conditions for the space to become a \(T_{1}\) space and Hausdorff space.


Residuated lattice \(\alpha \)-filter Co-annihilator Heyting algebra Prime \(\alpha \)-filter space 



This research was supported by a grant of National Natural Science Foundation of China (11571281).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.

Ethical approval

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

Informed consent

Informed consent was obtained from all individual participants included in the study.


  1. Bikhoff G (1973) Lattice theory. American Mathematical Society, ProvidenceGoogle Scholar
  2. Burris S, Sankappanavar HP (1981) A course in universal algebra. Springer, New YorkCrossRefzbMATHGoogle Scholar
  3. Grätzer G (1979) Lattice theory. W. H, Freeman and Company, San FranciscozbMATHGoogle Scholar
  4. Cornish WH (1973) Annulets and \(\alpha \)-ideals in distributive lattice. J Aust Math Soc 15:70–77MathSciNetCrossRefzbMATHGoogle Scholar
  5. Haveshki M, Mohamadhasani M (2015) On \(\alpha \)-filtetrs of \(BL\)-algebras. J Intell Fuzzy Syst 28:373–382zbMATHGoogle Scholar
  6. Hájek P (1998) Metamathematics of fuzzy logic. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  7. Liu LZ, Li KT (2007) Boolean filters and positive implicative filters of residuated lattices. Inf Sci 177:5725–5738MathSciNetCrossRefzbMATHGoogle Scholar
  8. Saeidk AB, Pourkhatoun M (2012) Obstinate filters in residuated lattices. Bull Math Soc Sci Math Roum Tome 103(55):413–422MathSciNetzbMATHGoogle Scholar
  9. Turunen E (1999) Mathematics behind fuzzy logic. Physica, HeidelbergzbMATHGoogle Scholar
  10. Ward M, Dilworth PR (1939) Residuated lattice. Trans Am Math Soc 45:335–354CrossRefGoogle Scholar
  11. Zhang XH, Shen JG (2006) On filters of residuated lattice. Chin QJ Math 21:443–447MathSciNetGoogle Scholar
  12. Zhang XH, Shen JG (2009) On lattice structure of filters in residuated lattices. Chin Q J Math 24(2):252–257MathSciNetzbMATHGoogle Scholar
  13. Zhu YQ, Xu Y (2010) On filter theory of residuated lattices. Inf Sci 180:3614–3632MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China

Personalised recommendations