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Finite Closed Sets of Functions in Multi-valued Logic

Received: 03 January 2017    Accepted: 14 January 2017    Published: 20 February 2017
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Abstract

The article is devoted to classification of close sets of functions in k-valued logic. We build the classification of finite closed sets. The sets contain only constants and unary functions since sets containing even a two-ary function are infinite. Formulas of the number of finite sets and minimal sets exist for all natural k. We find the numbers of closed sets containing the identity function f (x)=x or a constant for all k. We give the number of sets on levels of inclusions at k up to 10. The inclusion diagrams are present at k up to 5, at k=6 we give inclusion diagrams of sets containing only function f (x)=x and constant 0. We find isomorphic sets and use only one of the isomorphic sets to build the diagrams.

DOI 10.11648/j.pamj.20170601.13
Published in Pure and Applied Mathematics Journal (Volume 6, Issue 1, February 2017)
Page(s) 14-24
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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), 2024. Published by Science Publishing Group

Keywords

Discreat Mathematics, Multi-valued Logic, Function Classification

References
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  • Russian Research Center for Artificial Intelligence, Moscow, Russia

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    M. A. Malkov. (2017). Finite Closed Sets of Functions in Multi-valued Logic. Pure and Applied Mathematics Journal, 6(1), 14-24. https://doi.org/10.11648/j.pamj.20170601.13

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    M. A. Malkov. Finite Closed Sets of Functions in Multi-valued Logic. Pure Appl. Math. J. 2017, 6(1), 14-24. doi: 10.11648/j.pamj.20170601.13

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    M. A. Malkov. Finite Closed Sets of Functions in Multi-valued Logic. Pure Appl Math J. 2017;6(1):14-24. doi: 10.11648/j.pamj.20170601.13

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  • @article{10.11648/j.pamj.20170601.13,
      author = {M. A. Malkov},
      title = {Finite Closed Sets of Functions in Multi-valued Logic},
      journal = {Pure and Applied Mathematics Journal},
      volume = {6},
      number = {1},
      pages = {14-24},
      doi = {10.11648/j.pamj.20170601.13},
      url = {https://doi.org/10.11648/j.pamj.20170601.13},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20170601.13},
      abstract = {The article is devoted to classification of close sets of functions in k-valued logic. We build the classification of finite closed sets. The sets contain only constants and unary functions since sets containing even a two-ary function are infinite. Formulas of the number of finite sets and minimal sets exist for all natural k. We find the numbers of closed sets containing the identity function f (x)=x or a constant for all k. We give the number of sets on levels of inclusions at k up to 10. The inclusion diagrams are present at k up to 5, at k=6 we give inclusion diagrams of sets containing only function f (x)=x and constant 0. We find isomorphic sets and use only one of the isomorphic sets to build the diagrams.},
     year = {2017}
    }
    

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    AB  - The article is devoted to classification of close sets of functions in k-valued logic. We build the classification of finite closed sets. The sets contain only constants and unary functions since sets containing even a two-ary function are infinite. Formulas of the number of finite sets and minimal sets exist for all natural k. We find the numbers of closed sets containing the identity function f (x)=x or a constant for all k. We give the number of sets on levels of inclusions at k up to 10. The inclusion diagrams are present at k up to 5, at k=6 we give inclusion diagrams of sets containing only function f (x)=x and constant 0. We find isomorphic sets and use only one of the isomorphic sets to build the diagrams.
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