Skip to main content
SpringerLink
Log in

Nodal filters in hoop algebras

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

Abstract

Hoop algebras or hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach (Fundam Math 64:257–287, 1969, Fundam Math 69:1–14, 1970). In this paper, we introduce the notions of node and nodal filter in hoops and study some properties of them. First, we prove that the sets of all nodes are a bounded distributive lattice. Then by define some operations on \({\mathcal {NF}}(A)\), the set of all nodal filters in hoop A, we show that \({\mathcal {NF}}(A)\) is a Hertz algebra, Heyting algebra, Kleene algebra, semi-De Morgan algebra, Hilbert algebra and BCK-algebra. Finally, we investigate the relation among nodal filters and (positive) implicative, obstinate, prime and maximal filters in any hoops.

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

Similar content being viewed by others

References

  • Aglianó P, Ferreirim IM, Montagna F (2007) Basic hoops: an algebraic study of continuous t-norms. Stud Log 87:73–98

    Article  Google Scholar 

  • Balbes R, Horn A (1970) Injective and projective Heyting algebras. Trans Am Math Soc 148:549–559

    Article  MathSciNet  Google Scholar 

  • Bakhshi M (2016) Nodal filters in residuated lattices. J Intell Fuzzy Syst 30:2555–2562

    Article  Google Scholar 

  • Boicescu V, Filipoiu A, Georgescu G, Rudeanu S (1991) Łukasiewicz–Moisil algebras. Annals of discrete mathematics, vol 49. North-Holland, Amsterdam

    MATH  Google Scholar 

  • Blok WJ, Ferreirim IMA (2000) On the structure of hoops. Algebra Univers 43:233–257

    Article  MathSciNet  Google Scholar 

  • Borzooei RA, Aaly Kologani M (2014) Filter theory of hoop-algebras. J Adv Res Pure Math 6(4):72–86

    Article  MathSciNet  Google Scholar 

  • Borzooei RA, Aaly Kologani M (2015) Local and perfect semihoops. J Intell Fuzzy Syst 29:223–234

    Article  MathSciNet  Google Scholar 

  • Bosbach B (1969) Komplementäre Halbgruppen. Axiomatik und Arithmetik. Fundam Math 64:257–287

    Article  Google Scholar 

  • Bosbach B (1970) Komplementäre Halbgruppen. Kongruenzen und Quatienten. Fundam Math 69:1–14

    Article  Google Scholar 

  • Büchi JR, Owens TM (1975) Complemented monoids and hoops (unpublished manuscript)

  • Diego A (1966) Sur les algèbres de Hilbert. Collection de Logique Mathematique Série A 21:1–54

    MathSciNet  MATH  Google Scholar 

  • Georgescu G, Leustean L, Preoteasa V (2005) Pseudo-hoops. J Mult Valued Log Soft Comput 11(1–2):153–184

    MathSciNet  MATH  Google Scholar 

  • Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht

    Book  Google Scholar 

  • Kondo M (2011) Some types of filters in hoops. In: 41st IEEE international symposium on multiple-valued logic (ISMVL). IEEE, pp 50–53

  • Meng J, Jun YB (1994) BCK-algebra. Kyungmoonsa Co, Seoul

    Google Scholar 

  • Namdar A, Borzooei RA, Borumand Saeid A, Aaly Kologani M. Some results in Hoop algebras. J Intell Fuzzy Syst 32(3):1805–1813

    Article  Google Scholar 

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

    MATH  Google Scholar 

  • Porta H (1981) Sur quelques algèbres de la Logique. Port Math 40:210–238

    Google Scholar 

  • Sankappanavar HP (1987) Semi-De Morgan algebras. J Symb Log 52(3):712–724

    Article  MathSciNet  Google Scholar 

  • Tayebi Khorami R, Borumand Saeid A (2015a) New filters of BCK-algebras. Anal Univ Oradea Fasc Math Tom XXII(1):181–188

  • Tayebi Khorami R, Borumand Saeid A (2015b) Nodal filters of BL-algebras. J Intel Fuzzy Syst 28:1159–1167

  • Varlet JC (1973) Nodal filters in semilattices. Comnentationes Mathematicae Univer-sitatis Carolinae 14(2):263–277

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to express their appreciation for several excellent suggestions for improvements in this paper made by the editor and referees. Funding was provided by Shahid Beheshti University and Islamic Azad University Kerman.

Author information

Authors and Affiliations

  1. Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran

    A. Namdar

  2. Department of Mathematics, Shahid Beheshti University, Tehran, Iran

    R. A. Borzooei

Authors
  1. A. Namdar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. A. Borzooei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. A. Borzooei.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by A. Di Nola.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Namdar, A., Borzooei, R.A. Nodal filters in hoop algebras. Soft Comput 22, 7119–7128 (2018). https://doi.org/10.1007/s00500-017-2986-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-017-2986-8

Keywords

Search

Navigation