Skip to main content
SpringerLink
Log in

Isotone lattice-valued Boolean functions and cuts

  • Published:
Acta Scientiarum Mathematicarum Aims and scope Submit manuscript

Abstract

For an integer n ≥ 1, an n-ary lattice-valued Boolean function is a map from the n-th direct power of the 2-element Boolean lattice to a bounded lattice. In terms of closure systems and cuts, we characterize lattice-valued Boolean functions that can be given by linear combinations of elements of the co-domain lattice.

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

  1. N. N. Aǐzenberg, A. A. Bovdi, E. I. Gergo and F. É Geche, Algebraic aspects of threshold logic, Cybernetics 16 (1980), 188–193.

    Article  Google Scholar 

  2. D. Christofides, Influences of monotone Boolean functions, Discrete Mathematics 310 (2010), 1401–1402.

    Article  MathSciNet  Google Scholar 

  3. M. Couceiro and T. Waldhauser, Interpolation by polynomial functions of distributive lattices: a generalization of a theorem of R. L. Goodstein, Algebra Universalis 69 (2013), 287.299.

    Article  MathSciNet  Google Scholar 

  4. Y. Crama and P. L. Hammer, Boolean Functions, Theory, Algorithms, and Application, Cambridge University Press, 2010.

    Google Scholar 

  5. P. Crawley and P. Dilworth, Algebraic Theory of Lattices, Englewood Cliffs, N.J., Prentice-Hall, 1973.

    MATH  Google Scholar 

  6. B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1992.

    MATH  Google Scholar 

  7. R. L. Goodstein, The Solution of Equations in a Lattice, Proc. Roy. Soc. Edinburgh Sect. A 67 (1965/1967), 231–242.

    MathSciNet  MATH  Google Scholar 

  8. M. Grech, Regular symmetric groups of boolean functions, Discrete Mathematics 310 (2010), 2877–2882.

    Article  MathSciNet  Google Scholar 

  9. V. Gurvich and L. Khachiyan, On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions, Discrete Applied Mathematics 96–97 (1999), 363–373.

    Article  MathSciNet  Google Scholar 

  10. L. Hellerstein, On generalized constraints and certificates, Discrete Mathematics 226 (2001), 211–232.

    Article  MathSciNet  Google Scholar 

  11. E. K. Horvath, Invariance groups of threshold functions, Acta Cybernetica 11 (1994), 325–332.

    MathSciNet  MATH  Google Scholar 

  12. E. K. Horvath, B. Šešelja and A. Tepavčevič, Cardinality of height functions range in case of maximally many rectangular islands. computed by cuts, Central European J. Math. 11.2 (2013), 296–307.

    MathSciNet  MATH  Google Scholar 

  13. E. K. Horvath, B. Šešelja and A. Tepavčevič, A note on lattice variant of thresholdness of Boolean functions, manuscript available at http://www.math.u-szeged.hu/~horvath/mmnthreshold.pdf.

  14. N. Keller and H. Pilpel, Linear transformations of monotone functions on the discrete cube, Discrete Math. 309 (2009), 4210–4214.

    Article  MathSciNet  Google Scholar 

  15. D. Kleitman, On Dedekind’s problem: the number of monotone Boolean functions, Proc. Amer. Math. Soc. 21 (1969), 677–682.

    MathSciNet  MATH  Google Scholar 

  16. F. Lange and M. Grabisch, The interaction transform for functions on lattices, Discrete Math. 309 (2009), 4037–4048.

    Article  MathSciNet  Google Scholar 

  17. S. Muroga, Threshold logic and its applications, Wiley-Interscience, 1971.

    MATH  Google Scholar 

  18. S. Rudeanu, Boolean functions and equations, North-Holland, New York, 1974.

    MATH  Google Scholar 

  19. Ch. L. Sheng, Threshold logic, The Ryerson Press Toronto, Academic Press, London - New York, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bolyai Institute. Szeged, University of Szeged, Aradi vértanúk tere 1, Hungary, 6720

    Eszter K. Horváth

  2. Faculty of Science, University of Novi Sad, Trg D. Obradovića 4, 21000, Novi Sad, Serbia

    Branimir Šešelja & Andreja Tepavčevič

Authors
  1. Eszter K. Horváth
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Branimir Šešelja
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreja Tepavčevič
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Eszter K. Horváth, Branimir Šešelja or Andreja Tepavčevič.

Additional information

This paper is dedicated to Professor László Leindler on his 80th birthday

Communicated by M. Maróti

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Horváth, E.K., Šešelja, B. & Tepavčevič, A. Isotone lattice-valued Boolean functions and cuts. ActaSci.Math. 81, 375–380 (2015). https://doi.org/10.14232/actasm-014-331-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.14232/actasm-014-331-1

Key words and phrases

AMS Subject Classification

Search

Navigation