Galois and Post Algebras of Compositions (Superpositions)
Pure and Applied Mathematics Journal
Volume 6, Issue 4, August 2017, Pages: 114-119
Received: Jun. 10, 2017; Accepted: Jun. 22, 2017; Published: Jul. 20, 2017
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Author
Maydim Malkov, Russian Research Center for Artificial Intelligence, Moscow, Russia
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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.
Keywords
Function Algebra, Relation Algebra, Universal Post Algebra
To cite this article
Maydim Malkov, Galois and Post Algebras of Compositions (Superpositions), Pure and Applied Mathematics Journal. Vol. 6, No. 4, 2017, pp. 114-119. doi: 10.11648/j.pamj.20170604.12
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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