Skip to main content
SpringerLink
Log in

Tense operators on non-commutative residuated lattices

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

Abstract

In this paper, the algebraic properties of tense operators on a non-commutative residuated lattice are investigated. First, some examples and basic properties are given. Next, it is proved that the Dedekind–MacNeille completion of a tense non-commutative residuated lattice is again a tense non-commutative residuated lattice, together with suitable operations. In the sequel, the concept of tense filter is introduced and some structural theorems are given. Several characterizations of normal tense filters and maximal tense filters are obtained, as well. Also, some characterizations of tense non-commutative residuated lattices using normal tense filters are given.

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

Similar content being viewed by others

References

  • Bakhshi M (2015) Tense pseudo \({\sf MV}\)-algebras. In: Proceedings of the 4th Iranian joint congress on fuzzy and intelligent systems, September 9–11, 2015, pp 18–22

  • Bakhshi M (2014) Generalized fuzzy filters in non-commutative residuated lattices. Afr Mat 25:289–305

    Article  MathSciNet  MATH  Google Scholar 

  • Bakhshi M (2013) Some types of filters in non-commutative residuated lattices. Int J Math Comput 21:72–87

    MathSciNet  Google Scholar 

  • Blok WJ, Pigozzi D (1989) Algebrizable logics. Memoirs of the American Mathematical Society No. 396, American Mathematical Society, Providence

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

    Article  MathSciNet  MATH  Google Scholar 

  • Buşneag D, Piciu D, Paralescu J (2013) Divisible and semi-divisible residuated lattices. An Ştiinţ Univ Al I Cuza Iaşi Mat (NS) 14-45

  • Burges J (1984) Basic tense logic. In: Gabbay DM, Günther F (eds) Handbook of philosophical logic, vol II. Reidel, Dordrecht, pp 89–139

    Chapter  Google Scholar 

  • Botur M, Chajda I, Halaš R, Kolarřík M (2011) Tense operators on basic algebras. Int J Theor Phys 50:3737–3749

    Article  MathSciNet  MATH  Google Scholar 

  • Chajda I, Kolar̆ík M (2012) Dynamic effect algebras. Math Slovaca 62:379–388

    MathSciNet  Google Scholar 

  • Chajda I, Paseka J (2015) Tense operators in fuzzy logic. Fuzzy Sets Syst 276:100–113

    Article  MathSciNet  MATH  Google Scholar 

  • Chirită C (2011) Tense \(\theta \)-valued Łukasiewicz–Moisil algebras. J Mult Valued Log Soft Comput 17:1–24

    MATH  Google Scholar 

  • Ciungu LC (2006) Classes of residuated lattices. Ann Univ Craiova Math Comput Sci Ser 33:189–207

    MathSciNet  MATH  Google Scholar 

  • Diaconescu D, Georgescu G (2007) Tense operators on MV-algebras and Łukasiewicz–Moisil algebras. Fund Inform 81:379–408

    MathSciNet  MATH  Google Scholar 

  • Dilworth RP (1938) Non-commutative residuated lattices. Trans Am Math Soc 46:426–444

    Article  MathSciNet  MATH  Google Scholar 

  • Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics. Elsevier, Amsterdam

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Haveshki M, Saeid AB, Eslami E (2006) Some types of filters in \({\sf BL}\)-algebras. Soft Comput 10:657–664

    Article  MATH  Google Scholar 

  • Höhle U (1995) Commutative residuated monoids. In: Hohle U, Klement P (eds) Non-classical logics and their applications to fuzzy subsets. kluwer academic publishers, Dordrecht

  • Krull W (1924) Axiomatische Begründung der allgemeinen Ideal theorie. Sitzungsbrichte der physikalisch medizinischen Societäd der Erlangen 56:47–63

    MATH  Google Scholar 

  • Lianzhen L, Kaitai L (2007) Boolean filters and positive implicative filters of non-commutative residuated lattices. Inf Sci 177:5725–5738

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Pavelka J (1979) On fuzzy logic II. Enriched residuated lattices and semantics of propositional calcului. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 25:119–134

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Turunen E (1999) Mathematics behind fuzzy logic. Physica-Verlag, Wurzburg

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Ward M, Dilworth RP (1938) Residuated lattices. Proc Natl Acad Sci USA 24:162–164

    Article  MATH  Google Scholar 

Download references

Acknowledgments

I would like to express my sincere thanks to the referees for their valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bojnord, Bojnord, Iran

    Mahmood Bakhshi

Authors
  1. Mahmood Bakhshi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahmood Bakhshi.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

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

Bakhshi, M. Tense operators on non-commutative residuated lattices. Soft Comput 21, 4257–4268 (2017). https://doi.org/10.1007/s00500-016-2191-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-016-2191-1

Keywords

Search

Navigation