Constructions of Implications Satisfying the Order Property on a Complete Lattice
Automation, Control and Intelligent Systems
Volume 5, Issue 1, February 2017, Pages: 1-7
Received: Jan. 7, 2017; Accepted: Jan. 19, 2017; Published: Feb. 23, 2017
Views 2830      Downloads 91
Authors
Yuan Wang, College of Information Science and Technology, Yancheng Teachers University, Yancheng, People's Republic of China
Keming Tang, College of Information Science and Technology, Yancheng Teachers University, Yancheng, People's Republic of China
Zhudeng Wang, School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, People's Republic of China
Article Tools
Follow on us
Abstract
In this paper, we further investigate the constructions of fuzzy connectives on a complete lattice. We firstly illustrate the concepts of left (right) semi-uninorms and implications satisfying the order property by means of some examples. Then we give out the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation.
Keywords
Fuzzy Logic, Fuzzy Connective, Implication, Order Property
To cite this article
Yuan Wang, Keming Tang, Zhudeng Wang, Constructions of Implications Satisfying the Order Property on a Complete Lattice, Automation, Control and Intelligent Systems. Vol. 5, No. 1, 2017, pp. 1-7. doi: 10.11648/j.acis.20170501.11
Copyright
Copyright © 2017 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]
J. Fodor and M. Roubens, “Fuzzy Preference Modelling and Multicriteria Decision Support”, Theory and Decision Library, Series D: System Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, 1994.
[2]
G. J. Klir and B. Yuan, “Fuzzy Sets and Fuzzy Logic, Theory and Applications”, Prentice Hall, New Jersey, 1995.
[3]
E. P. Klement, R. Mesiar and E. Pap, “Triangular Norms”, Trends in Logic-Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.
[4]
M. Baczynski and B. Jayaram, “Fuzzy Implication”, Studies in Fuzziness and Soft Computing, Vol. 231, Springer, Berlin, 2008.
[5]
M. Baczynski and B. Jayaram, “QL-implications: some properties and intersections”, Fuzzy Sets and Systems, 161, 158-188, 2010.
[6]
H. Bustince, P. Burillo and F. Soria, “Automorphisms, negations and implication operators”, Fuzzy Sets and Systems, 134, 209-229, 2003.
[7]
F. Durante, E. P. Klement, R. Mesiar and C. Sempi, “Conjunctors and their residual implicators: characterizations and construction methods”, Mediterranean Journal of Mathematics, 4, 343-356, 2007.
[8]
Y. Shi, B. Van Gasse, D. Ruan and E. E. Kerre, “On dependencies and independencies of fuzzy implication axioms”, Fuzzy Sets and Systems, 161, 1388-1405, 2010.
[9]
J. Fodor and T. Keresztfalvi, “Nonstandard conjunctions and implications in fuzzy logic”, International Journal of Approximate Reasoning, 12, 69-84, 1995.
[10]
J. Fodor, “Srict preference relations based on weak t-norms”, Fuzzy Sets and Systems, 43, 327-336, 1991.
[11]
Z. D. Wang and Y. D. Yu, “Pseudo-t-norms and implication operators on a complete Brouwerian lattice”, Fuzzy Sets and Systems, 132, 113-124, 2002.
[12]
Y. Su and Z. D. Wang, “Pseudo-uninorms and coimplications on a complete lattice”, Fuzzy Sets and Systems, 224, 53-62, 2013.
[13]
Z. D. Wang and J. X. Fang, “Residual operators of left and right uninorms on a complete lattice”, Fuzzy Sets and Systems, 160, 22-31, 2009.
[14]
H. W. Liu, “Semi-uninorm and implications on a complete lattice”, Fuzzy Sets and Systems, 191, 72-82, 2012.
[15]
Y. Ouyang, “On fuzzy implications determined by aggregation operators”, Information Sciences, 193, 153-162, 2012.
[16]
R. R. Yager and A. Rybalov, “Uninorm aggregation operators”, Fuzzy Sets and Systems, 80, 111-120, 1996.
[17]
J. Fodor, R. R. Yager and A. Rybalov, “Structure of uninorms”, Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems, 5, 411-427, 1997.
[18]
M. Mas, M. Monserrat and J. Torrens, “On left and right uninorms”, Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems, 9, 491-507, 2001.
[19]
M. Mas, M. Monserrat and J. Torrens, “On left and right uninorms on a finite chain”, Fuzzy Sets and Systems, 146, 3-17, 2004.
[20]
Z. D. Wang and J. X. Fang, “Residual coimplicators of left and right uninorms on a complete lattice”, Fuzzy Sets and Systems, 160, 2086-2096, 2009.
[21]
Y. Su, Z. D. Wang and K. M. Tang, “Left and right semi-uninorms on a complete lattice”, Kybernetika, 49, 948-961, 2013.
[22]
B. De Baets and J. Fodor, “Residual operators of uninorms”, Soft Computing, 3, 89-100, 1999.
[23]
M. Mas, M. Monserrat and J. Torrens, “Two types of implications derived from uninorms”, Fuzzy Sets and Systems, 158, 2612-2626, 2007.
[24]
Z. D. Wang, “Generating pseudo-t-norms and implication operators”, Fuzzy Sets and Systems, 157, 398-410, 2006.
[25]
Y. Su and Z. D. Wang, “Constructing implications and coimplications on a complete lattice”, Fuzzy Sets and Systems, 247, 68-80, 2014.
[26]
X. Y. Hao, M. X. Niu and Z. D. Wang, “The relations between implications and left (right) semi-uninorms on a complete lattice”, Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems, 23, 245-261, 2015.
[27]
X. Y. Hao, M. X. Niu, Y. Wang and Z. D. Wang, “Constructing conjunctive left (right) semi-uninorms and implications satisfying the neutrality principle”, Journal of Intelligent and Fuzzy Systems, 31, 1819-1829, 2016.
[28]
M. X. Niu, X. Y. Hao and Z. D. Wang, “Relations among implications, coimplications and left (right) semi-uninorms”, Journal of Intelligent and Fuzzy Systems, 29, 927-938, 2015.
[29]
Z. D. Wang, M. X. Niu and X. Y. Hao, “Constructions of coimplications and left (right) semi-uninorms on a complete lattice”, Information Sciences, 317, 181-195, 2015.
[30]
Z. D. Wang, “Left (right) semi-uninorms and coimplications on a complete lattice”, Fuzzy Sets and Systems, 287, 227-239, 2016.
[31]
G. Birkhoff, “Lattice Theory”, American Mathematical Society Colloquium Publishers, Providence, 1967.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186