| Peer-Reviewed

Some Relationships between Left (Right) Semi-Uninorms and Implications on a Complete Lattice

Received: 4 August 2014     Accepted: 18 August 2014     Published: 30 August 2014
Views:       Downloads:
Abstract

In this paper, we study the relationships between left (right) semi-uninorms and implications on a complete lattice. We firstly discuss the residual operations of left and right semi-uninorms and show that the right (left) residual operator of a conjunctive right (left) ∨-distributive left (right) semi-uninorm is a right ∧-distributive implication that satisfies the neutrality principle. Then, we investigate the left and right semi-uninorms induced by an implication, give some conditions such that two operations induced by an implication constitute left or right semi-uninorms, and demonstrate that the operations induced by a right ∧-distributive implication, which satisfies the order property or neutrality principle, are left (right) ∨-distributive left (right) semi-uninorms or right (left) semi-uninorms. Finally, we reveal the relationships between conjunctive right (left) ∨-distributive left (right) semi-uninorms and right ∧-distributive implications which satisfy the neutrality principle.

Published in Automation, Control and Intelligent Systems (Volume 2, Issue 3)
DOI 10.11648/j.acis.20140203.12
Page(s) 33-41
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), 2014. Published by Science Publishing Group

Keywords

Fuzzy Logic, Fuzzy Connective, Left (Right) Semi-Uninorm, Implication, Neutrality Principle, Order Property

References
[1] E. P. Klement, R. Mesiar, and E. Pap, “Triangular Norms”, Trends in Logic-Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.
[2] P. Flondor, G. Georgescu, and A. orgulescu, “Pseudo-t-norms and pseudo-BL-algebras”, Soft Computing, 5, 355-371, 2001.
[3] J. Fdor and T. Keresztfalvi, “Nonstandard conjunctions and implications in fuzzy logic”, International Journal of Approximate Reasoning, 12, 69-84, 1995.
[4] J. Fdor, “Srict preference relations based on weak t-norms”, Fuzzy Sets and Systems, 43, 327-336, 1991.
[5] 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.
[6] R. R. Yager and A. Rybalov, “Uninorm aggregation operators”, Fuzzy Sets and Systems, 80, 111-120, 1996.
[7] J. Fodor, R. R. Yager, and A. Rybalov, “Structure of uninorms”, Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems, 5, 411-427, 1997.
[8] D. Gabbay and G. Metcalfe, “fuzzy logics based on [0,1)-continuous uninorms”, Arch. Math. Logic, 46, 425-449, 2007.
[9] A. K. Tsadiras and K. G. Margaritis, “The MYCIN certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons”, Fuzzy Sets and Systems, 93, 263-274, 1998.
[10] R. R. Yager, “Uninorms in fuzzy system modeling”, Fuzzy Sets and Systems, 122, 167-175, 2001.
[11] R. R. Yager, “Defending against strategic manipulation in uninorm-based multi-agent decision making”, European J. Oper. Res., 141, 217-232, 2002.
[12] M. Mas, M. Monserrat, and J. Torrens, “On left and right uninorms”, Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems, 9, 491-507, 2001.
[13] M. Mas, M. Monserrat, and J. Torrens, “On left and right uninorms on a finite chain”, Fuzzy Sets and Systems, 146, 3-17, 2004.
[14] 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.
[15] 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.
[16] H. W. Liu, “Semi-uninorm and implications on a complete lattice”, Fuzzy Sets and Systems, 191, 72-82, 2012.
[17] Y. Su, Z. D. Wang, and K. M. Tang, “Left and right semi-uninorms on a complete lattice”, Kybernetika, 49, 948-961, 2013.
[18] B. De Baets and J. Fodor, “Residual operators of uninorms”, Soft Computing, 3, 89-100, 1999.
[19] M. Mas, M. Monserrat, and J. Torrens, “Two types of implications derived from uninorms”, Fuzzy Sets and Systems, 158, 2612-2626, 2007.
[20] D. Ruiz and J. Torrens, “Residual implications and co-implications from idempotent uninorms”, Kybernetika, 40, 21-38, 2004.
[21] G. Birkhoff, “Lattice Theory”, American Mathematical Society Colloquium Publishers, Providence, 1967.
[22] F. Suarez Garcia and P. Gil Alvarez, “Two families of fuzzy intergrals”, Fuzzy Sets and Systems, 18, 67-81, 1986.
[23] B. Bassan and F. Spizzichino, “Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes”, J. Multivariate Anal., 93, 313-339, 2005.
[24] F. Durante, E. P. Klement, and R. Mesiar et al., “Conjunctors and their residual implicators: characterizations and construct methods”, Mediterranean J. Math., 4, 343-356, 2007.
[25] G. De Cooman and E. E. Kerre, “Order norms on bounded partially ordered sets”, J. Fuzzy Math., 2, 281-310, 1994.
[26] B. De Baets, “Idempotent uninorms”, European J. Oper. Res., 118, 631-642, 1999.
[27] M. Baczynski and B. Jayaram, “Fuzzy Implication”, Studies in Fuzziness and Soft Computing, Vol. 231, Springer, Berlin, 2008.
[28] B. De Baets, “Coimplicators, the forgotten connectives”, Tatra Mountains Math. Publ., 12, 229-240, 1997.
[29] Y. Su and Z. D. Wang, “Pseudo-uninorms and coimplications on a complete lattice”, Fuzzy Sets and Systems, 224, 53-62, 2013.
Cite This Article
  • APA Style

    Yuan Wang, Keming Tang, Zhudeng Wang. (2014). Some Relationships between Left (Right) Semi-Uninorms and Implications on a Complete Lattice. Automation, Control and Intelligent Systems, 2(3), 33-41. https://doi.org/10.11648/j.acis.20140203.12

    Copy | Download

    ACS Style

    Yuan Wang; Keming Tang; Zhudeng Wang. Some Relationships between Left (Right) Semi-Uninorms and Implications on a Complete Lattice. Autom. Control Intell. Syst. 2014, 2(3), 33-41. doi: 10.11648/j.acis.20140203.12

    Copy | Download

    AMA Style

    Yuan Wang, Keming Tang, Zhudeng Wang. Some Relationships between Left (Right) Semi-Uninorms and Implications on a Complete Lattice. Autom Control Intell Syst. 2014;2(3):33-41. doi: 10.11648/j.acis.20140203.12

    Copy | Download

  • @article{10.11648/j.acis.20140203.12,
      author = {Yuan Wang and Keming Tang and Zhudeng Wang},
      title = {Some Relationships between Left (Right) Semi-Uninorms and Implications on a Complete Lattice},
      journal = {Automation, Control and Intelligent Systems},
      volume = {2},
      number = {3},
      pages = {33-41},
      doi = {10.11648/j.acis.20140203.12},
      url = {https://doi.org/10.11648/j.acis.20140203.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acis.20140203.12},
      abstract = {In this paper, we study the relationships between left (right) semi-uninorms and implications on a complete lattice. We firstly discuss the residual operations of left and right semi-uninorms and show that the right (left) residual operator of a conjunctive right (left) ∨-distributive left (right) semi-uninorm is a right ∧-distributive implication that satisfies the neutrality principle. Then, we investigate the left and right semi-uninorms induced by an implication, give some conditions such that two operations induced by an implication constitute left or right semi-uninorms, and demonstrate that the operations induced by a right ∧-distributive implication, which satisfies the order property or neutrality principle, are left (right) ∨-distributive left (right) semi-uninorms or right (left) semi-uninorms. Finally, we reveal the relationships between conjunctive right (left) ∨-distributive left (right) semi-uninorms and right ∧-distributive implications which satisfy the neutrality principle.},
     year = {2014}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Some Relationships between Left (Right) Semi-Uninorms and Implications on a Complete Lattice
    AU  - Yuan Wang
    AU  - Keming Tang
    AU  - Zhudeng Wang
    Y1  - 2014/08/30
    PY  - 2014
    N1  - https://doi.org/10.11648/j.acis.20140203.12
    DO  - 10.11648/j.acis.20140203.12
    T2  - Automation, Control and Intelligent Systems
    JF  - Automation, Control and Intelligent Systems
    JO  - Automation, Control and Intelligent Systems
    SP  - 33
    EP  - 41
    PB  - Science Publishing Group
    SN  - 2328-5591
    UR  - https://doi.org/10.11648/j.acis.20140203.12
    AB  - In this paper, we study the relationships between left (right) semi-uninorms and implications on a complete lattice. We firstly discuss the residual operations of left and right semi-uninorms and show that the right (left) residual operator of a conjunctive right (left) ∨-distributive left (right) semi-uninorm is a right ∧-distributive implication that satisfies the neutrality principle. Then, we investigate the left and right semi-uninorms induced by an implication, give some conditions such that two operations induced by an implication constitute left or right semi-uninorms, and demonstrate that the operations induced by a right ∧-distributive implication, which satisfies the order property or neutrality principle, are left (right) ∨-distributive left (right) semi-uninorms or right (left) semi-uninorms. Finally, we reveal the relationships between conjunctive right (left) ∨-distributive left (right) semi-uninorms and right ∧-distributive implications which satisfy the neutrality principle.
    VL  - 2
    IS  - 3
    ER  - 

    Copy | Download

Author Information
  • College of Information Science and Technology, Yancheng Teachers University, Yancheng 224002, People's Republic of China

  • College of Information Science and Technology, Yancheng Teachers University, Yancheng 224002, People's Republic of China

  • School of Mathematical Sciences, Yancheng Teachers University, Jiangsu 224002, People's Republic of China

  • Sections