Skip to main content
SpringerLink
Log in

Involutive right-residuated l-groupoids

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

Abstract

A common generalization of orthomodular lattices and residuated lattices is provided corresponding to bounded lattices with an involution and sectionally extensive mappings. It turns out that such a generalization can be based on integral right-residuated l-groupoids. This general framework is applied to MV-algebras, orthomodular lattices, Nelson algebras, basic algebras and Heyting algebras.

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

  • Bĕlohlávek R (2003) Some properties of residuated lattices. Czechoslovak Math J 53(128):161–171

    Article  MathSciNet  MATH  Google Scholar 

  • Bĕlohlávek R (2002) Fuzzy relational systems. Foundations and Principles, Kluwer, New-York

    Book  MATH  Google Scholar 

  • Blyth TS, Janowitz MF (1972) Residuation theory. International Series of Monographs in Pure and Applied Mathematics, vol 102. Pergamon Press, Oxford

  • Botur M, Chajda I, Halaš R (2010) Are basic algebras residuated structures? Soft Comp 14:251–255

    Article  MATH  Google Scholar 

  • Chajda I (1998) Locally regular varieties. Acta Sci Math (Szeged) 64:431–435

    MathSciNet  MATH  Google Scholar 

  • Chajda I (2003) An extension of relative pseudocomplemention to non-distributive lattices. Acta Sci Math (Szeged) 69:491–496

  • Chajda I (2011) Basic algebras and their applications, an overview. Contributions to General algebra. Verlag J Heyn 20:1–10

  • Chajda I, Halaš R, Kűhr J (2005) Distributive lattices with sectionally antitone involutions. Acta Sci Math (Szeged) 71:19–33

    MathSciNet  MATH  Google Scholar 

  • Chajda I, Halaš R, Kűhr J (2009) Many-valued quantum algebras. Algebra Univ 60:63–90

    Article  MATH  Google Scholar 

  • Chajda I, Kühr J (2013) Ideals and congruences of basic algebras. Soft Comp 17:401–410

    Article  MATH  Google Scholar 

  • Chajda I, Radeleczki S (2003) On varieties defined by pseudocomplemented nondistributive lattices. Publ Math Debrecen 63(4):737–750

    MathSciNet  MATH  Google Scholar 

  • Chajda I, Radeleczki S (2014) An approach to orthomodular lattices via lattices with an antitone involution. Math Slovaca, [accepted for publication]

  • Cignoli R (1986) The class of Kleene algebras satisfying an interpolation property and Nelson algebras. Algebra Univ 23:262–292

    Article  MathSciNet  MATH  Google Scholar 

  • Cornelis C, De Cock M, Radzikowska AM (2007) Vaguely quantified rough sets. Rough sets, fuzzy sets, aata mining and granular computing. Springer, Berlin-Heidelberg, pp 87–94

    Chapter  Google Scholar 

  • Csákány B (1970) Characterisation of regular varieties. Acta Sci Math (Szeged) 31:187–189

    MathSciNet  MATH  Google Scholar 

  • Czelakowski J (2008) Additivity of the commutator and residuation. Rep Math Logic 43:109–132

    MathSciNet  MATH  Google Scholar 

  • Dilworth RP, Ward M (1939) Residuated lattices. Trans Amer Math Soc 45:335–354

    Article  MathSciNet  Google Scholar 

  • Järvinen J, Pagliani P, Radeleczki S (2013) Information completeness in Nelson algebras of rough sets induced by quasiorders. Studia Logica 1015:1073–1092

    Article  Google Scholar 

  • Jarvinen J, Radeleczki S (2014) Monteiro spaces and rough sets determined by quasiorder relations: models for Nelson algebras. Fund Inform 131(2):205–215

    MathSciNet  Google Scholar 

  • Kondo M (2011) Modal operators on commutative residuated lattices. Math Slovaca 611:1–14

    Article  Google Scholar 

  • Pagliani P, Chakraborty M (2008) A geometry of approximation. Rough set theory: logic, algebra and topology of conceptual patterns. Springer, New York

    Book  Google Scholar 

  • Spinks M, Veroff R (2010) Constructive logic with strong negation as a substructural logic. J Logic Comp 20(4):761–793

    Article  Google Scholar 

Download references

Acknowledgments

We thank the anonymous referee for his/her valuable comments and hints which made our paper more readable. We also express our thanks to Stephane Foldes for his helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Algebra and Geometry, Palacky University, 17. listopadu 12, Olomouc, 771 46, Czech Republic

    Ivan Chajda

  2. Mathematical Institute, University of Miskolc, 3515, Miskolc-Egyetemvaros, Hungary

    Sándor Radeleczki

Authors
  1. Ivan Chajda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sándor Radeleczki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sándor Radeleczki.

Additional information

Communicated by A. Di Nola.

The work of the first author is supported by the Project I 1923-N25 by Austrian Science Found (FWF), and Czech Grant Agency (GACR).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chajda, I., Radeleczki, S. Involutive right-residuated l-groupoids. Soft Comput 20, 119–131 (2016). https://doi.org/10.1007/s00500-015-1765-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-015-1765-7

Keywords

Mathematics Subject Classification

Search

Navigation