# Implications, coimplications and left semi-uninorms on a complete lattice

Document Type: Research Paper

Authors

1 College of Information Engineering, Yancheng Teachers University, Jiangsu 224002, People's Republic of China

2 School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, People's Republic of China

Abstract

In this paper, we firstly show that the \$N\$-dual operation of the right residual implication, which is induced by a left-conjunctive right arbitrary \$\vee\$-distributive left semi-uninorm, is the right residual coimplication induced by its \$N\$-dual operation. As a dual result, the \$N\$-dual operation of the right residual coimplication, which is induced by a left-disjunctive right arbitrary \$\wedge\$-distributive left semi-uninorm, is the right residual implication induced by its \$N\$-dual operation. Then, we demonstrate that the \$N\$-dual operations of the left semi-uninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left semi-uninorms. Finally, we reveal the relationships between conjunctive right arbitrary \$\vee\$-distributive left semi-uninorms induced by implications and disjunctive right arbitrary \$\wedge\$-distributive left semi-uninorms induced by coimplications, where both implications and coimplications satisfy the neutrality principle.

Keywords

### References

[1] M. Baczynski and B. Jayaram, Fuzzy implications, Studies in Fuzziness and Soft Computing,
Springer, Berlin, 231 (2008).
[2] S. Burris and H. P. Sankappanavar, A course in universal algebra, World Publishing Corporation,
Beijing 1981.
[3] B. De Baets, Coimplicators, the forgotten connectives, Tatra Mountains Mathematical Publications,
12 (1997), 229{240.
[4] B. De Baets and J. Fodor, Residual operators of uninorms, Soft Computing, 3 (1999), 89{100.
[5] F. Durante, E. P. Klement, R. Mesiar and C. Sempi, Conjunctors and their residual impli-
cators: characterizations and construction methods, Mediterranean Journal of Mathematics,
4 (2007), 343{356.
[6] P. Flondor, G. Georgescu and A. Lorgulescu, Pseudo-t-norms and pseudo-BL-algebras, Soft
Computing, 5 (2001), 355{371.
[7] J. Fodor and T. Keresztfalvi, Nonstandard conjunctions and implications in fuzzy logic,
International Journal of Approximate Reasoning, 12 (1995), 69{84.
[8] J. C. 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.
[9] J. Fodor, R. R. Yager and A. Rybalov, Structure of uninorms, International Journal of
Uncertainty, Fuzziness and Knowledge-Based Systems, 5 (1997), 411{427.
[10] D. Gabbay and G. Metcalfe, Fuzzy logics based on [0; 1)-continuous uninorms, Archive for
Mathematical Logic, 46 (2007), 425{449.
[11] G. Gratzer, Lattice theory: foundation, Birkhauser, Springer Basel AG, 2011.
[12] S. Jenei and F. Montagna, A general method for constructing left-continuous t-norms, Fuzzy
Sets and Systems, 136 (2003), 263{282.
[13] H. W. Liu, Semi-uninorm and implications on a complete lattice, Fuzzy Sets and Systems,
191 (2012), 72{82.
[14] Z. Ma and W. M.Wu, Logical operators on complete lattices, Information Sicences, 55 (1991),
77{97.
[15] M. Mas, M. Monserrat and J. Torrens, On left and right uninorms, International Journal of
Uncertainty, Fuzziness and Knowledge-Based Systems, 9 (2001), 491{507.
[16] M. Mas, M. Monserrat and J. Torrens, On left and right uninorms on a nite chain, Fuzzy
Sets and Systems, 146 (2004), 3{17.
[17] M. Mas, M. Monserrat and J. Torrens, Two types of implications derived from uninorms,
Fuzzy Sets and Systems, 158 (2007), 2612{2626.

[18] D. Ruiz and J. Torrens, Residual implications and co-implications from idempotent uninorms,
Kybernetika, 40 (2004), 21{38.
[19] Y. Su and H. W. Liu, Characterizations of residual coimplications of pseudo-uninorms on a
complete lattice, Fuzzy Sets and Systems, 261 (2015), 44{59.
[20] Y. Su and Z. D. Wang, Pseudo-uninorms and coimplications on a complete lattice, Fuzzy
Sets and Systems, 224 (2013), 53{62.
[21] Y. Su and Z. D. Wang, Constructing implications and coimplications on a complete lattice,
Fuzzy Sets and Systems, 247 (2014), 68{80.
[22] Y. Su, Z. D. Wang and K. M. Tang, Left and right semi-uninorms on a complete lattice,
Kybernetika, 49 (2013), 948{961.
[23] Z. D. Wang and J. X. Fang, Residual operators of left and right uninorms on a complete
lattice, Fuzzy Sets and Systems, 160 (2009), 22{31.
[24] Z. D. Wang and J. X. Fang, Residual coimplicators of left and right uninorms on a complete
lattice, Fuzzy Sets and Systems, 160 (2009), 2086{2096.
[25] R. R. Yager, Uninorms in fuzzy system modeling, Fuzzy Sets and Systems, 122 (2001),
167{175.
[26] R. R. Yager, Defending against strategic manipulation in uninorm-based multi-agent decision
making, European Journal of Operational Research, 141 (2002), 217{232.
[27] R. R. Yager and A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems, 80
(1996), 111{120.