Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions

Maydim Malkov

doi:doi:10.11648/j.pamj.20190804.11

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Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions

Maydim Malkov

Published in Pure and Applied Mathematics Journal (Volume 8, Issue 4)

Received: 25 July 2019 Accepted: 15 August 2019 Published: 3 September 2019

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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.

Published in	Pure and Applied Mathematics Journal (Volume 8, Issue 4)
DOI	10.11648/j.pamj.20190804.11
Page(s)	72-76
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), 2019. Published by Science Publishing Group

Keywords

Algebra of Real Functions, Algebra of Superpositions, Computable Real Functions, Computable Real Numbers

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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Maydim Malkov. (2019). Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions. Pure and Applied Mathematics Journal, 8(4), 72-76. https://doi.org/10.11648/j.pamj.20190804.11

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Maydim Malkov. Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions. Pure Appl. Math. J. 2019, 8(4), 72-76. doi: 10.11648/j.pamj.20190804.11

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Maydim Malkov. Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions. Pure Appl Math J. 2019;8(4):72-76. doi: 10.11648/j.pamj.20190804.11

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@article{10.11648/j.pamj.20190804.11,
  author = {Maydim Malkov},
  title = {Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {8},
  number = {4},
  pages = {72-76},
  doi = {10.11648/j.pamj.20190804.11},
  url = {https://doi.org/10.11648/j.pamj.20190804.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20190804.11},
  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.},
 year = {2019}
}

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TY  - JOUR
T1  - Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions
AU  - Maydim Malkov
Y1  - 2019/09/03
PY  - 2019
N1  - https://doi.org/10.11648/j.pamj.20190804.11
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JO  - Pure and Applied Mathematics Journal
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PB  - Science Publishing Group
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AB  - 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.
VL  - 8
IS  - 4
ER  -

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Author Information

Maydim Malkov

Department of Mathematics, Russian Research Center for Artificial Intelligence, Moscow, Russia

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APA Style

Maydim Malkov. (2019). Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions. Pure and Applied Mathematics Journal, 8(4), 72-76. https://doi.org/10.11648/j.pamj.20190804.11

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ACS Style

Maydim Malkov. Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions. Pure Appl. Math. J. 2019, 8(4), 72-76. doi: 10.11648/j.pamj.20190804.11

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AMA Style

Maydim Malkov. Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions. Pure Appl Math J. 2019;8(4):72-76. doi: 10.11648/j.pamj.20190804.11

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@article{10.11648/j.pamj.20190804.11,
  author = {Maydim Malkov},
  title = {Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {8},
  number = {4},
  pages = {72-76},
  doi = {10.11648/j.pamj.20190804.11},
  url = {https://doi.org/10.11648/j.pamj.20190804.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20190804.11},
  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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions
AU  - Maydim Malkov
Y1  - 2019/09/03
PY  - 2019
N1  - https://doi.org/10.11648/j.pamj.20190804.11
DO  - 10.11648/j.pamj.20190804.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 72
EP  - 76
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20190804.11
AB  - 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.
VL  - 8
IS  - 4
ER  -

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