Function as the Generator of Parametric T-norms

Md. Shohel Babu; Shifat Ahmed

doi:doi:10.11648/j.ajam.20170504.13

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Function as the Generator of Parametric T-norms

Md. Shohel Babu, Shifat Ahmed

Published in American Journal of Applied Mathematics (Volume 5, Issue 4)

Received: 23 May 2017 Accepted: 21 June 2017 Published: 24 July 2017

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Abstract

The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (usually, addition or multiplication) into a T-norm. In order to allow using non-bijective generators, which do not have the inverse function, we have used the notion of pseudo-inverse function. Many families of related t-norms can be defined by an explicit formula depending on a parameter p. Firstly; some continuous and decreasing parametric functions have been selected. Then generate parametric T-norms by using those functions based on additive generator.

Published in	American Journal of Applied Mathematics (Volume 5, Issue 4)
DOI	10.11648/j.ajam.20170504.13
Page(s)	114-118
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), 2017. Published by Science Publishing Group

Keywords

Pseudo-inverse, Additive generators, Parametric T-norms, Yager’s Product ,Dombi’s Product , Aczel-Alsina ,Frank Product , Schweizer and Sklar

References

[1]	Md. Shohel Babu, Dr. Abeda Sultana, Md. Abdul Alim, Continuous Functions as the Generators of T-norms, IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 2 Ver. I (Mar - Apr. 2015), PP 35-38.
[2]	Shohel Babu, Fatema Tuj Johora, Abdul Alim, Investigation of Order among Some Known T-norms, American Journal of Applied Mathematics 2015; 3 (5): 229-232 Published online September 25, 2015 doi: 10.11648/j.ajam.20150305.14 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online).
[3]	George J Klir, Yuan Bo, Fuzzy Sets And Fuzzy Logic, Theory And Applications, Prentice-Hall Inc. N. J. U.S.A. 1995.
[4]	Mirko Navara (2007), “Triangular Norms And Conforms” Scholarpedia.
[5]	Peter J Crickmore, Fuzzy Sets And System, Centre For Environmental Investigation Inc.
[6]	Peter Vicenik, A Note On Generators Of T-Norms; Department Of Mathematics, Slovak Technical University, Radlinskeho 11, 813 68 Bratislava, Slovak Republic.
[7]	Didier Dobois, Prade Henri, FUZZY SET AND SYSTEM, THEORY AND APPLICATIONS, Academic press INC, New York.
[8]	Lowen, FUZZYSET THEORY, Department of Mathematics and Computer Science, University of Antwerp; Belgium, Basic Concepts, Techniques and Bibliography, Kluwer Academic Publishers Dordeecht/Boston/London.
[9]	Matteo Bianchi, The logic of the strongest and the weakest tnorms, Fuzzy Sets Syst. 276 (2015) 31–42, http://dx.doi.org/10.1016/j.fss.2015.01. 13.
[10]	Wladyslaw Homenda, TRIANGULAR NORMS, UNI-AND NULLNORMS, BALANCED NORMS, THE CASES OF THE HIERACHY OF ITERATIVE OPERATORS, Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland.
[11]	Mirta N. K., FUZZY SET THEORY RELATIONAL STRUCTURE USING T-NORMS AND MATHLAB, Department of Mathematics, University of Dhaka.
[12]	Klement, Erich Perer; Mesiar, Radko; and Pap, Endre (2000), Triangular Norms. Dordrecht: Kluwer. ISBN 0-7923-6416-3.

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Md. Shohel Babu, Shifat Ahmed. (2017). Function as the Generator of Parametric T-norms. American Journal of Applied Mathematics, 5(4), 114-118. https://doi.org/10.11648/j.ajam.20170504.13

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ACS Style

Md. Shohel Babu; Shifat Ahmed. Function as the Generator of Parametric T-norms. Am. J. Appl. Math. 2017, 5(4), 114-118. doi: 10.11648/j.ajam.20170504.13

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AMA Style

Md. Shohel Babu, Shifat Ahmed. Function as the Generator of Parametric T-norms. Am J Appl Math. 2017;5(4):114-118. doi: 10.11648/j.ajam.20170504.13

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@article{10.11648/j.ajam.20170504.13,
  author = {Md. Shohel Babu and Shifat Ahmed},
  title = {Function as the Generator of Parametric T-norms},
  journal = {American Journal of Applied Mathematics},
  volume = {5},
  number = {4},
  pages = {114-118},
  doi = {10.11648/j.ajam.20170504.13},
  url = {https://doi.org/10.11648/j.ajam.20170504.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20170504.13},
  abstract = {The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (usually, addition or multiplication) into a T-norm. In order to allow using non-bijective generators, which do not have the inverse function, we have used the notion of pseudo-inverse function. Many families of related t-norms can be defined by an explicit formula depending on a parameter p. Firstly; some continuous and decreasing parametric functions have been selected. Then generate parametric T-norms by using those functions based on additive generator.},
 year = {2017}
}

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TY  - JOUR
T1  - Function as the Generator of Parametric T-norms
AU  - Md. Shohel Babu
AU  - Shifat Ahmed
Y1  - 2017/07/24
PY  - 2017
N1  - https://doi.org/10.11648/j.ajam.20170504.13
DO  - 10.11648/j.ajam.20170504.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 114
EP  - 118
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20170504.13
AB  - The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (usually, addition or multiplication) into a T-norm. In order to allow using non-bijective generators, which do not have the inverse function, we have used the notion of pseudo-inverse function. Many families of related t-norms can be defined by an explicit formula depending on a parameter p. Firstly; some continuous and decreasing parametric functions have been selected. Then generate parametric T-norms by using those functions based on additive generator.
VL  - 5
IS  - 4
ER  -

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Author Information

Md. Shohel Babu

Computer Science & Engineering, Southeast University, Dhaka, Bangladesh
Shifat Ahmed

Electrical & Electronic Engineering, Southeast University, Dhaka, Bangladesh

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APA Style

Md. Shohel Babu, Shifat Ahmed. (2017). Function as the Generator of Parametric T-norms. American Journal of Applied Mathematics, 5(4), 114-118. https://doi.org/10.11648/j.ajam.20170504.13

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ACS Style

Md. Shohel Babu; Shifat Ahmed. Function as the Generator of Parametric T-norms. Am. J. Appl. Math. 2017, 5(4), 114-118. doi: 10.11648/j.ajam.20170504.13

Copy | Download

AMA Style

Md. Shohel Babu, Shifat Ahmed. Function as the Generator of Parametric T-norms. Am J Appl Math. 2017;5(4):114-118. doi: 10.11648/j.ajam.20170504.13

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@article{10.11648/j.ajam.20170504.13,
  author = {Md. Shohel Babu and Shifat Ahmed},
  title = {Function as the Generator of Parametric T-norms},
  journal = {American Journal of Applied Mathematics},
  volume = {5},
  number = {4},
  pages = {114-118},
  doi = {10.11648/j.ajam.20170504.13},
  url = {https://doi.org/10.11648/j.ajam.20170504.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20170504.13},
  abstract = {The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (usually, addition or multiplication) into a T-norm. In order to allow using non-bijective generators, which do not have the inverse function, we have used the notion of pseudo-inverse function. Many families of related t-norms can be defined by an explicit formula depending on a parameter p. Firstly; some continuous and decreasing parametric functions have been selected. Then generate parametric T-norms by using those functions based on additive generator.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Function as the Generator of Parametric T-norms
AU  - Md. Shohel Babu
AU  - Shifat Ahmed
Y1  - 2017/07/24
PY  - 2017
N1  - https://doi.org/10.11648/j.ajam.20170504.13
DO  - 10.11648/j.ajam.20170504.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 114
EP  - 118
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20170504.13
AB  - The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (usually, addition or multiplication) into a T-norm. In order to allow using non-bijective generators, which do not have the inverse function, we have used the notion of pseudo-inverse function. Many families of related t-norms can be defined by an explicit formula depending on a parameter p. Firstly; some continuous and decreasing parametric functions have been selected. Then generate parametric T-norms by using those functions based on additive generator.
VL  - 5
IS  - 4
ER  -

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