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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
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
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
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