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Finite Closed Sets of Functions in Multi-valued Logic

Received: 3 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.

Published in Pure and Applied Mathematics Journal (Volume 6, Issue 1)
DOI 10.11648/j.pamj.20170601.13
Page(s) 14-24
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

Discreat Mathematics, Multi-valued Logic, Function Classification

References
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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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    ACS Style

    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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    AMA Style

    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://article.sciencepublishinggroup.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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    N1  - https://doi.org/10.11648/j.pamj.20170601.13
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    UR  - https://doi.org/10.11648/j.pamj.20170601.13
    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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    IS  - 1
    ER  - 

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  • Russian Research Center for Artificial Intelligence, Moscow, Russia

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