Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions
Pure and Applied Mathematics Journal
Volume 8, Issue 4, August 2019, Pages: 72-76
Received: Jul. 25, 2019; Accepted: Aug. 15, 2019; Published: Sep. 3, 2019
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Author
Maydim Malkov, Department of Mathematics, Russian Research Center for Artificial Intelligence, Moscow, Russia
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Abstract
Real numbers are divided into fictitious (non-computable) and essential (computable). Fictitious numbers do not have numerical values, essential numbers have algorithms for constructing these numbers with any exactness. The set of fictitious numbers is continual, the set of essential numbers is countable. Functions are also divided into fictitious, defined over the set of fictitious numbers, and essential, defined over the set of essential numbers. Essential functions have an algorithm for calculating any value with any exactness. All functions of applied mathematics and some functions of abstract mathematics are essential The set these functions is countable. The four upper levels of classification of real functions are constructed. This classification uses superpositions of functions and diagonal sets borrowed from the algebra of finite-valued functions.
Keywords
Algebra of Real Functions, Algebra of Superpositions, Computable Real Functions, Computable Real Numbers
To cite this article
Maydim Malkov, Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions, Pure and Applied Mathematics Journal. Vol. 8, No. 4, 2019, pp. 72-76. doi: 10.11648/j.pamj.20190804.11
Copyright
Copyright © 2019 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.
References
[1]
E. L. Post, Two-valued iterative systems of mathematical logic, Princeton, Princeton Univ. Press (1941).
[2]
M. A. Malkov, Logic algebra and Post algebra (theory of two-valued functions) (Russian), Moscow, Mathematical logic (2012).
[3]
D. Lau, Functions algebra on finite sets, Berlin, Springer (2006).
[4]
A. I. Malcev, I. A. Malcev, Iterative Post algebras, Moscow, Nauka (2012).
[5]
Ju. I. Janov, A. A. Muchnik, On existence of k-valued closed classes without finite basis (Russian), Dokl. Acad. Nauk SSSR, (1), 44-46 (1959).
[6]
M. A. Malkov, Classification of closed sets of functions in multi-valued logic, SOP Transactions on applied Math., (1: 3), 96-105 (2014).
[7]
A. V. Kuznetsov, On means for detecting of non-deductibility and inexpressibleness (Russian), in Logical conclusion, Moscow, Nauka, 5-33 (1979).
[8]
S. S. Marchenkov, On FE-precomplete classes of countable logic (Russian), Discrete Mathematics, (28: 2), 51-57 (2016).
[9]
S. V. Yablonsky, Functional constructions in k-valued logic (Russian), Proceedings of Mat. Institute of the USSR Academy of Sciences. V. A. Steklova, (51) 5-142 (1958).
[10]
I. G. Rosenberg, Über die functionale vollständigkeit in dem mehrvertigen logiken von mehreren verändlichen auf endlichen mengen, Rozpravy Cs. Academic Ved. Ser. Math. Nat. Sci., (80) 3-93 (1970).
[11]
M. Malkov, Algebra of finite-valued functions: Classification of functions and subalgebras, essential and fictitious subalgebras, Pure and Applied Math. J., (8: 2) 30-36 (2019).
[12]
G. Rousseau, Completeness in finite algebras with a single operation, Proc. Amer. Math. Soc., (18), 1009-1013.
[13]
P. Schofield, Independent conditions for completeness of finite algebras, J. London Math. Soc., (44) 413-423 (1969).
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