Finite Closed Sets of Functions in Multi-valued Logic

M. A. Malkov

doi:doi:10.11648/j.pamj.20170601.13

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Finite Closed Sets of Functions in Multi-valued Logic

M. A. Malkov

Published in Pure and Applied Mathematics Journal (Volume 6, Issue 1)

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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APA Style

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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TY  - JOUR
T1  - Finite Closed Sets of Functions in Multi-valued Logic
AU  - M. A. Malkov
Y1  - 2017/02/20
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170601.13
DO  - 10.11648/j.pamj.20170601.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 14
EP  - 24
PB  - Science Publishing Group
SN  - 2326-9812
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.
VL  - 6
IS  - 1
ER  -

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Author Information

M. A. Malkov

Russian Research Center for Artificial Intelligence, Moscow, Russia

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APA Style

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Finite Closed Sets of Functions in Multi-valued Logic
AU  - M. A. Malkov
Y1  - 2017/02/20
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170601.13
DO  - 10.11648/j.pamj.20170601.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 14
EP  - 24
PB  - Science Publishing Group
SN  - 2326-9812
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.
VL  - 6
IS  - 1
ER  -

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