Volume 7, Issue 6, December 2018, Page: 95-100
Post and Jablonsky Algebras of Compositions (Superpositions)
Maydim Malkov, Departament of Mathematics, Research Center for Artificial Intelligence, Moscow, Russia
Received: Nov. 29, 2018;       Accepted: Dec. 14, 2018;       Published: Jan. 10, 2019
DOI: 10.11648/j.pamj.20180706.13      View  227      Downloads  75
Abstract
There are two algebras of compositions, Post and Jablonsky algebras. Definitions of these algebras was very simple. The article gives mathematically precise definition of these algebras by using Mal’cev’s definitions of the algebras. A. I. Mal’cev defined pre-iterative and iterative algebras of compositions. The significant extension of pre-iterative algebra is given in the article. Iterative algebra is incorrect. E. L. Post used implicitly pre-iterative algebra. S. V. Jablonsky used implicitly iterative algebra. The Jablonsky algebra has the operation of adding fictitious variables. But this operation is not primitive, since the addition of fictitious variables is possible at absence of this operation. If fictitious functions are deleted in the Jablonsky algebra then this algebra becomes correct. A natural classification of closed sets is given and fictitious closed sets are exposed. The number of fictitious closed sets is continual, the number of essential closed sets is countable.
Keywords
Post Algebras, Closed Sets of Functions and Relations, Logic of Superpositions
To cite this article
Maydim Malkov, Post and Jablonsky Algebras of Compositions (Superpositions), Pure and Applied Mathematics Journal. Vol. 7, No. 6, 2018, pp. 95-100. doi: 10.11648/j.pamj.20180706.13
Copyright
Copyright © 2018 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.
Reference
[1]
E. L. Post Introduction to a general theory of elementary propositions Amer. J. Math. 43:4 163–185 (1921).
[2]
E. L. Post Two-valued iterative systems of mathematical logic Princeton Princeton Univ. Press (1941).
[3]
S. V. Jablonsky, G. P. Gavrilov, V. B. Kudryavcev Functions of algebra logic and Post classes (Russian) M. Nauka (1966).
[4]
A. I. Mal’cev Post iterative algebras (Russian) Novosibirsk NGU (1976).
[5]
D. Lau Functions algebras on finite sets N. Y. Springer (2006).
[6]
M. A. Malkov Poat’s thesis and wrong Yanov-Muchnik's statement in multi-valued logic Pure and Appl. Math. J. 4:4 172-177 (2015).
[7]
M. A. Malkov Galois and Post Algebras of Compositions (Superpositions) Pure and Appl. Math. J. 6:4 2017 114-119
[8]
Y. I. Yanov, A. A. Muchnik On existence of k-valued closed classes without finite bases (Russean) Doklady AN SSSR 127: 1 44–46 (1959).
[9]
M. A. Malkov Classification of closed sets of functions in multi-valued logic SOP Transaction on Appl. Math. 1: 3 96–105 (2014).
[10]
M. A. Malkov Classification of Boolean functions and their closed sets SOP Transaction on Appl. Math. 1:2 172–193 (2014).
Browse journals by subject