Skip to main content
Springer Nature Link
Log in

On congruences in weak implicative semi-lattices

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

Abstract

In this paper we characterize the congruences in classes of algebras that properly include varieties of interest for the logic. These algebras are obtained by weakening the main features of Heyting algebras \((H, \wedge , \vee , \rightarrow ,0, 1)\) but retaining most of their algebraic consequences. The residuation property \(z\le x\rightarrow y\) if and only if \(x \wedge z \le y\) is replaced by the condition: if \(z\le x\rightarrow y\) then \(x \wedge z \le y\). Some of further algebraic conditions of Heyting algebras are required.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Abad M, Cornejo JM, Díaz Varela JP (2012) Free-decomposability in varieties of semi-Heyting algebras. Math Logic Q 58(3):168–176

    Article  MathSciNet  MATH  Google Scholar 

  • Abad M, Cornejo JM, JP Díaz Varela (2013) Semi-Heyting algebras term-equivalent to Gödel algebras. Order 30(2):625–642

    Article  MathSciNet  MATH  Google Scholar 

  • Abad M, Cornejo JM, Díaz Varela JP (2010) The variety generated by semi-Heyting chains. Soft Comput 15(4):721–728

    Article  MATH  Google Scholar 

  • Abad M, Cornejo JM, Díaz Varela JP (2011) The variety of semi-Heyting algebras satisfying the equation \((0\rightarrow 1)^{*} \vee (0\rightarrow 1)^{**} \approx 1\). Rep Math Logic 46:75–90

    MATH  Google Scholar 

  • Balbes R, Dwinger P (1974) Distributive lattices. University of Missouri Press, Columbia

    MATH  Google Scholar 

  • Caicedo X, Cignoli R (2001) An algebraic approach to intuitionistic connectives. J Symb Log 66(4):1620–1636

    Article  MathSciNet  MATH  Google Scholar 

  • Celani SA, Jansana R (2005) Bounded distributive lattices with strict implication. Math Logic Q 51(3):219–246

    Article  MathSciNet  MATH  Google Scholar 

  • Cornejo JM (2011) Semi-intuitionistic logic. Stud Log 98(1–2):9–25

    Article  MathSciNet  MATH  Google Scholar 

  • Curry HB (1963) Foundations of mathematical logic. McGraw-Hill, New York

    MATH  Google Scholar 

  • Ertola R, San Martín HJ (2011) On some compatible operations on Heyting algebras. Stud Log 98(3):331–345

    Article  MathSciNet  MATH  Google Scholar 

  • Fitting MC (1969) Intuitionistic logic, model theory and forcing. Amsterdam, North-Holland

    MATH  Google Scholar 

  • Grätzer G (1962) On boolean functions (Notes on lattice theory II). Revue Roumaine de Mathematiques Pures et Appliquees 7:693–697

    MathSciNet  MATH  Google Scholar 

  • Kaarli K, Pixley AF (2001) Polynomial completeness in algebraic systems. Chapman and Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Nemitz W (1965) Implicative semi-lattices. Trans Am Math Soc 117:128–142

    Article  MathSciNet  MATH  Google Scholar 

  • Pixley A (1972) Completeness in arithmetical algebras. Algebra Universalis 2(1):179–196

    Article  MathSciNet  MATH  Google Scholar 

  • Rasiowa H (1974) An algebraic approach to non-classical logics. Amsterdam, North Holland

    MATH  Google Scholar 

  • Rasiowa H, Sikorski R (1970) The mathematics of metamathematics, 3rd edn. Polish Scientific Publishers, Warsaw

  • Sankappanavar HP (2007) Semi-Heyting algebras: an abstraction from Heyting algebras. Actas del IX Congreso Antonio Monteiro, Bahía Blanca

    MATH  Google Scholar 

  • Sankappanavar HP (2011) Expansions of semi-Heyting algebras I: discriminator varieties. Stud Log 98(1–2):27–81

    Article  MathSciNet  MATH  Google Scholar 

  • San Martín HJ (2013) Compatible operations in some subvarieties of the variety of weak Heyting algebras. In: 8th conference of the European society for fuzzy logic and technology (EUSFLAT 2013), pp 475–480

Download references

Acknowledgments

This work was supported by Grants CONICET PIP 112-201101-00636 and UNLP 11/X667.

Author information

Authors and Affiliations

  1. Departamento de Matemática, CONICET and Facultad de Ciencias Exactas, UNLP, Casilla de correos 172, 1900, La Plata, Argentina

    Hernán Javier San Martín

Authors
  1. Hernán Javier San Martín
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hernán Javier San Martín.

Ethics declarations

Conflict of interest

The author Hernán Javier San Martín declares that there is 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

San Martín, H.J. On congruences in weak implicative semi-lattices. Soft Comput 21, 3167–3176 (2017). https://doi.org/10.1007/s00500-016-2188-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-016-2188-9

Keywords

Search

Navigation