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Clones of Self-Dual and Self-K-Al Functions in K-valued Logic

Received: 17 February 2017     Accepted: 24 February 2017     Published: 10 March 2017
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Abstract

We give a classification of dual functions, they are m-al functions. We call a function m-al with respect to an operator if the operator lives any function unchanged after m times of using the operator. And 2 ≤ m ≤ k. Functions with different m have very different properties. We give theoretical results for clones of self-dual (m = 2) and self- -al (m = k) functions in k-valued logic at k ≤ 3. And we give numerical results for clones of self-dual and self-3-al functions in 3-valued logic. In particular, the inclusion graphs of clones of self-dual and of self-3-al functions are not a lattice.

Published in Pure and Applied Mathematics Journal (Volume 6, Issue 2)
DOI 10.11648/j.pamj.20170602.11
Page(s) 59-70
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Discreate Mathematics, K-Valued Function Algebra, Selfdual Functions

References
[1] Malkov M. A., Classifications of closed sets of functions in multi-valued logic, SOP transactions on applied math. 1:3, 96-105 (2014), http://www.scipublish.com/journals/AM/papers/download/2102-970.pdf.
[2] Yablonskiy S. V.: Functional constructions in many-valued logics (Russian). Tr. Mat. Inst. Steklova, 515-142 (1958).
[3] Lau D., Functions algebra on finite sets, Springer (2006).
[4] Csàkàny B., All minimal clones on the three-element set, Acta Cybernet., 6, 227-238 (1983).
[5] Marchenkov S. S., Demetrovics J., Hannak L., On closed classes of self-dual functions in P3. (Russian) Metody Diskretn. Anal. 34, 38-73 (1980).
[6] Machida H., On closed sets of three-valued monotone logical functions. In: Colloquia Mathematica Societatis Janos Bolyai 28, Finite Algebra and multiple-valued logic, Szeged (Hungary), 441-467 (1979).
[7] Post E. L., The two-valued iterative systems of mathematical logic. Princeton Univ. Press, Princeton (1941).
[8] Mal’cev A. I., Iterative Post algebras, NGU, Novosibirsk, (Russian) (1976).
[9] Rosenberg, I. G., Űber die Verschiedenheit maximaler Klassen in Pk. Rev. Roumaine Math. Pures Appl. 14, 431–438 (1969).
[10] Malkov M. A. Classifications of Boolean Functions and Their Closed Sets, SOP transactions on applied math. 1:2, 172-193 (2014), http://www.scipublish.com/journals/AM/papers/download/2102-522.pdf.
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  • APA Style

    M. A. Malkov. (2017). Clones of Self-Dual and Self-K-Al Functions in K-valued Logic. Pure and Applied Mathematics Journal, 6(2), 59-70. https://doi.org/10.11648/j.pamj.20170602.11

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    ACS Style

    M. A. Malkov. Clones of Self-Dual and Self-K-Al Functions in K-valued Logic. Pure Appl. Math. J. 2017, 6(2), 59-70. doi: 10.11648/j.pamj.20170602.11

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    AMA Style

    M. A. Malkov. Clones of Self-Dual and Self-K-Al Functions in K-valued Logic. Pure Appl Math J. 2017;6(2):59-70. doi: 10.11648/j.pamj.20170602.11

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  • @article{10.11648/j.pamj.20170602.11,
      author = {M. A. Malkov},
      title = {Clones of Self-Dual and Self-K-Al Functions in K-valued Logic},
      journal = {Pure and Applied Mathematics Journal},
      volume = {6},
      number = {2},
      pages = {59-70},
      doi = {10.11648/j.pamj.20170602.11},
      url = {https://doi.org/10.11648/j.pamj.20170602.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170602.11},
      abstract = {We give a classification of dual functions, they are m-al functions. We call a function m-al with respect to an operator if the operator lives any function unchanged after m times of using the operator. And 2 ≤ m ≤ k. Functions with different m have very different properties. We give theoretical results for clones of self-dual (m = 2) and self- -al (m = k) functions in k-valued logic at k ≤ 3. And we give numerical results for clones of self-dual and self-3-al functions in 3-valued logic. In particular, the inclusion graphs of clones of self-dual and of self-3-al functions are not a lattice.},
     year = {2017}
    }
    

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Author Information
  • Russian Research Center for Artificial Intelligence, Moscow, Russia

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