Prominent GE-Filters and GE-Morphisms in GE-Algebras

Abstract

The relationship between a transitive GE-algebra and a belligerent GE-algebra (also, between an antisymmetric GE-algebra and a left exchangeable GE-algebra) is displayed. A condition for the trivial GE-filter to be a belligerent GE-filter is provided. The least GE-filter containing a given GE-filter and one element is formed. Conditions under which any set can be turned into a GE-filter are described. The notions of \(\odot \)-GE-algebras and prominent GE-filters are introduced, and their properties are investigated. The relationship between a prominent GE-filter and a GE-filter are considered, and conditions for a GE-filter and the trivial GE-filter to be a prominent GE-filter are given. The conditions under which the upset of an element will be a prominent GE-filter are examined, and the extension property for the prominent GE-filter is established. The notion of GE-morphism is introduced, and the GE-morphism theorem is considered. Conditions for the kernel of a GE-morphism to be a belligerent GE-filter are provided, and the condition that if the kernel of a GE-morphism is a belligerent GE-filter, then the trivial GE-filter becomes a belligerent GE-filter is given. A condition for the kernel of a GE-morphism to be a belligerent GE-filter is discussed, and the vice versa is also considered.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Bandaru, R.K., Borumand Saeid, A., Jun, Y.B.: On GE-algebras. Bull Sect Logic (2020). https://doi.org/10.18778/0138-0680.2020.20. (in press)

    Article  Google Scholar 

  2. 2.

    Bandaru RK, Borumand Saeid A, Jun YB (2020) Belligerent GE-filter in GE-algebras (submitted)

  3. 3.

    Borumand Saeid A, Rezaei A, Bandaru RK, Jun YB (2020) Voluntary GE-filters and further results of GE-filters in GE-algebras (submitted)

  4. 4.

    Borzooei, R.A., Shohani, J.: On generalized Hilbert algebras. Ital J Pure Appl Math 29, 71–86 (2012)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Chajda, I., Halas, R.: Congruences and ideal as in Hilbert algebras. Kyungpook Math J 39, 429–432 (1999)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Chajda, I., Halas, R., Jun, Y.B.: Annihilators and deductive systems in commutative Hilbert algebras. Commentationes Mathematicae Universitatis Carolinae 43(3), 407–417 (2002)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Jun, Y.B.: Commutative Hilbert algebras, Soochow. J Math 22(4), 477–484 (1996)

    MathSciNet  Google Scholar 

  8. 8.

    Jun, Y.B., Kim, K.H.: H-filters of Hilbert algebras. Scientiae Mathematicae Japonicae e–2005, 231–236 (2005)

    MathSciNet  Google Scholar 

  9. 9.

    Soleimani Nasab, A., Borumand Saeid, A.: Semi maximal filter in Hilbert algebra. J Intell Fuzzy Syst 30(1), 7–15 (2016). https://doi.org/10.3233/IFS-151706

    Article  MATH  Google Scholar 

  10. 10.

    Soleimani Nasab, A., Borumand Saeid, A.: Stonean Hilbert algebra. J Intell Fuzzy Syst 30(1), 485–492 (2016). https://doi.org/10.3233/IFS-151773

    Article  MATH  Google Scholar 

  11. 11.

    Soleimani Nasab A, Borumand Saeid A. Study of Hilbert algebras in point of filters. Analele Stiintifice ale Universitatii “vidius” Constanta. 2016;24(2):221–251. https://doi.org/10.1515/auom-2016-0039.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ravikumar Bandaru.

Additional information

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

Verify currency and authenticity via CrossMark

Cite this article

Rezaei, A., Bandaru, R., Saeid, A.B. et al. Prominent GE-Filters and GE-Morphisms in GE-Algebras. Afr. Mat. (2021). https://doi.org/10.1007/s13370-021-00886-6

Download citation

Keywords

  • Commutative
  • Transitive
  • Left exchangeable
  • Antisymmetric
  • GE-algebra
  • GE-filter
  • Prominent GE-filter

Mathematics Subject Classification

  • 03G25c
  • 06F35