Galois and Post Algebras of Compositions (Superpositions)

Maydim Malkov

doi:doi:10.11648/j.pamj.20170604.12

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Galois and Post Algebras of Compositions (Superpositions)

Maydim Malkov

Published in Pure and Applied Mathematics Journal (Volume 6, Issue 4)

Received: 10 June 2017 Accepted: 22 June 2017 Published: 20 July 2017

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Abstract

The Galois algebra and the universal Post algebra of compositions are constructed. The universe of the Galois algebra contains relations, both discrete and continuous. The found proofs of Galois connections are shorter and simpler. It is noted that anti-isomorphism of the two algebras of functions and of relations allows to transfer the results of the modern algebra of functions to the algebra of relations, and vice versa, to transfer the results of the modern algebra of relations to the algebra of functions. A new Post algebra is constructed by using pre-iterative algebra and by adding relations as one more universe of the algebra. The universes of relations and functions are discrete or continuous. It is proved that the Post algebra of relations and the Galois algebra are equal. This allows to replace the operation of conjunction by the operation of substitution and to exclude the operation of exist quantifier.

Published in	Pure and Applied Mathematics Journal (Volume 6, Issue 4)
DOI	10.11648/j.pamj.20170604.12
Page(s)	114-119
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

Function Algebra, Relation Algebra, Universal Post Algebra

References

[1]	Post E. L., Two-valued iterative systems of mathematical logic. Princeton Univ. Press, Princeton (1941).
[2]	Rosenbloom Paul C., Post algebras. I. Postulates and general theory, Amer. J. Math. 64, 167-–188 (1942).
[3]	Mal’cev A. I., Iterative Post algebras, NGU, Novosibirsk, (Russian) (1976).
[4]	Jablonskij S. V., Functional constructions in many-valued logics (Russian). Tr. Mat. Inst. Steklova, 51 5-142 (1958).
[5]	Jablonskij, S. W., Gavrilov, G. P., Kudryavcev, V. B., Boolean function and Post classes (Russian), Nauka, Moscow (1966).
[6]	Birkgoff G., Lattice theory, American Math. Soc., Providence Rhode island (1940)
[7]	Geiger D., Closed systems of functions and predicates Pacific J. Math. 27, 95-–100 (1968).
[8]	Bondarchuk, V. G., Kaluzhnin, L. A., Kotov, V. N., Romov, B. A., Galois theory for Post algebras I–II. (Russian) Kibernetika, 3, 1–-10 (1969), 5, 1-–9 (1969). English translation: Cybernetics, 243–-252 and 531-–539 (1969).
[9]	Marchenkov C., Basic of Boolean functions theory (Russian), Fizmatlit, Moscow (2000).
[10]	Malkov M. A., Classification of closed sets of functions in multi-valued logic. Sop transaction on applied math. 1:3 (2014).
[11]	Lau D., Functions algebra on finite sets, Springer (2006).

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Maydim Malkov. (2017). Galois and Post Algebras of Compositions (Superpositions). Pure and Applied Mathematics Journal, 6(4), 114-119. https://doi.org/10.11648/j.pamj.20170604.12

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ACS Style

Maydim Malkov. Galois and Post Algebras of Compositions (Superpositions). Pure Appl. Math. J. 2017, 6(4), 114-119. doi: 10.11648/j.pamj.20170604.12

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AMA Style

Maydim Malkov. Galois and Post Algebras of Compositions (Superpositions). Pure Appl Math J. 2017;6(4):114-119. doi: 10.11648/j.pamj.20170604.12

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@article{10.11648/j.pamj.20170604.12,
  author = {Maydim Malkov},
  title = {Galois and Post Algebras of Compositions (Superpositions)},
  journal = {Pure and Applied Mathematics Journal},
  volume = {6},
  number = {4},
  pages = {114-119},
  doi = {10.11648/j.pamj.20170604.12},
  url = {https://doi.org/10.11648/j.pamj.20170604.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170604.12},
  abstract = {The Galois algebra and the universal Post algebra of compositions are constructed. The universe of the Galois algebra contains relations, both discrete and continuous. The found proofs of Galois connections are shorter and simpler. It is noted that anti-isomorphism of the two algebras of functions and of relations allows to transfer the results of the modern algebra of functions to the algebra of relations, and vice versa, to transfer the results of the modern algebra of relations to the algebra of functions. A new Post algebra is constructed by using pre-iterative algebra and by adding relations as one more universe of the algebra. The universes of relations and functions are discrete or continuous. It is proved that the Post algebra of relations and the Galois algebra are equal. This allows to replace the operation of conjunction by the operation of substitution and to exclude the operation of exist quantifier.},
 year = {2017}
}

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TY  - JOUR
T1  - Galois and Post Algebras of Compositions (Superpositions)
AU  - Maydim Malkov
Y1  - 2017/07/20
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170604.12
DO  - 10.11648/j.pamj.20170604.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 114
EP  - 119
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20170604.12
AB  - The Galois algebra and the universal Post algebra of compositions are constructed. The universe of the Galois algebra contains relations, both discrete and continuous. The found proofs of Galois connections are shorter and simpler. It is noted that anti-isomorphism of the two algebras of functions and of relations allows to transfer the results of the modern algebra of functions to the algebra of relations, and vice versa, to transfer the results of the modern algebra of relations to the algebra of functions. A new Post algebra is constructed by using pre-iterative algebra and by adding relations as one more universe of the algebra. The universes of relations and functions are discrete or continuous. It is proved that the Post algebra of relations and the Galois algebra are equal. This allows to replace the operation of conjunction by the operation of substitution and to exclude the operation of exist quantifier.
VL  - 6
IS  - 4
ER  -

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Author Information

Maydim Malkov

Russian Research Center for Artificial Intelligence, Moscow, Russia

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Plain Text BibTeX RIS

APA Style

Maydim Malkov. (2017). Galois and Post Algebras of Compositions (Superpositions). Pure and Applied Mathematics Journal, 6(4), 114-119. https://doi.org/10.11648/j.pamj.20170604.12

Copy | Download

ACS Style

Maydim Malkov. Galois and Post Algebras of Compositions (Superpositions). Pure Appl. Math. J. 2017, 6(4), 114-119. doi: 10.11648/j.pamj.20170604.12

Copy | Download

AMA Style

Maydim Malkov. Galois and Post Algebras of Compositions (Superpositions). Pure Appl Math J. 2017;6(4):114-119. doi: 10.11648/j.pamj.20170604.12

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@article{10.11648/j.pamj.20170604.12,
  author = {Maydim Malkov},
  title = {Galois and Post Algebras of Compositions (Superpositions)},
  journal = {Pure and Applied Mathematics Journal},
  volume = {6},
  number = {4},
  pages = {114-119},
  doi = {10.11648/j.pamj.20170604.12},
  url = {https://doi.org/10.11648/j.pamj.20170604.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170604.12},
  abstract = {The Galois algebra and the universal Post algebra of compositions are constructed. The universe of the Galois algebra contains relations, both discrete and continuous. The found proofs of Galois connections are shorter and simpler. It is noted that anti-isomorphism of the two algebras of functions and of relations allows to transfer the results of the modern algebra of functions to the algebra of relations, and vice versa, to transfer the results of the modern algebra of relations to the algebra of functions. A new Post algebra is constructed by using pre-iterative algebra and by adding relations as one more universe of the algebra. The universes of relations and functions are discrete or continuous. It is proved that the Post algebra of relations and the Galois algebra are equal. This allows to replace the operation of conjunction by the operation of substitution and to exclude the operation of exist quantifier.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Galois and Post Algebras of Compositions (Superpositions)
AU  - Maydim Malkov
Y1  - 2017/07/20
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170604.12
DO  - 10.11648/j.pamj.20170604.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 114
EP  - 119
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20170604.12
AB  - The Galois algebra and the universal Post algebra of compositions are constructed. The universe of the Galois algebra contains relations, both discrete and continuous. The found proofs of Galois connections are shorter and simpler. It is noted that anti-isomorphism of the two algebras of functions and of relations allows to transfer the results of the modern algebra of functions to the algebra of relations, and vice versa, to transfer the results of the modern algebra of relations to the algebra of functions. A new Post algebra is constructed by using pre-iterative algebra and by adding relations as one more universe of the algebra. The universes of relations and functions are discrete or continuous. It is proved that the Post algebra of relations and the Galois algebra are equal. This allows to replace the operation of conjunction by the operation of substitution and to exclude the operation of exist quantifier.
VL  - 6
IS  - 4
ER  -

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