Abstract:

Uninorms are important generalizations of triangular norms and conorms, with a neutral element lying anywhere in the unit interval, and left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms. In this paper, we firstly introduce the concepts of left and right semi-uninorms on a complete lattice and illustrate these notions by means of some examples. Then, we lay bare the formulas for calculating the upper and lower approximation left (right) semi-uninorms of a binary operation. Finally, we discuss the relations between the upper approximation left (right) semi-uninorms of a given binary operation and the lower approximation left (right) semi-uninorms of its dual operation.

Keywords:

uninorm, fuzzy connective, left (right) semi-uninorm, upper (lower) approximation

03B52, 03E72

References:

1. B. De Baets: Coimplicators, the forgotten connectives. Tatra Mountains Math. Publ. 12 (1997), 229-240.   CrossRef
2. B. De Baets: Idempotent uninorms. European J. Oper. Res. 118 (1999), 631-642.   CrossRef
3. B. De Baets and J. Fodor: Van Melle's combining function in MYCIN is a representable uninorm: an alternative proof. Fuzzy Sets and Systems 104 (1999), 133-136.   CrossRef
4. B. Bassan and F. Spizzichino: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93 (2005), 313-339.   CrossRef
5. G. Birkhoff: Lattice Theory. American Mathematical Society Colloquium Publishers, Providence 1967.   CrossRef
6. S. Burris and H. P. Sankappanavar: A Course in Universal Algebra. Springer-Verlag, New York 1981.   CrossRef
7. G. De Cooman and E. E. Kerre: Order norms on bounded partially ordered sets. J. Fuzzy Math. 2 (1994), 281-310.   CrossRef
8. F. Durante, E. P. Klement and R. Mesiar et al.: Conjunctors and their residual implicators: characterizations and construct methods. Mediterranean J. Math. 4 (2007), 343-356.   CrossRef
9. J. Fodor, R. R. Yager and A. Rybalov: Structure of uninorms. Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems 5 (1997), 411-427.   CrossRef
10. D. Gabbay and G. Metcalfe: fuzzy logics based on $[0,1)$-continuous uninorms. Arch. Math. Logic 46 (2007), 425-449.   CrossRef
11. S. Gottwald: A Treatise on Many-Valued Logics. Studies in Logic and Computation Vol. 9, Research Studies Press, Baldock 2001.   CrossRef
12. S. Jenei: A characterization theorem on the rotation construction for triangular norms. Fuzzy Sets and Systems 136 (2003), 283-289.   CrossRef
13. S. Jenei: How to construct left-continuous triangular norms-state of the art. Fuzzy Sets and Systems 143 (2004), 27-45.   CrossRef
14. S. Jenei and F. Montagna: A general method for constructing left-continuous $t$-norms. Fuzzy Sets and Systems 136 (2003), 263-282.   CrossRef
15. H. W. Liu: Semi-uninorm and implications on a complete lattice. Fuzzy Sets and Systems 191 (2012), 72-82.   CrossRef
16. Z. Ma and W. M. Wu: Logical operators on complete lattices. Inform. Sci. 55 (1991), 77-97.   CrossRef
17. M. Mas, M. Monserrat and J. Torrens: On left and right uninorms. Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems 9 (2001), 491-507.   CrossRef
18. M. Mas, M. Monserrat and J. Torrens: On left and right uninorms on a finite chain. Fuzzy Sets and Systems 146 (2004), 3-17.   CrossRef
19. M. Mas, M. Monserrat and J. Torrens: Two types of implications derived from uninorms. Fuzzy Sets and Systems 158 (2007), 2612-2626.   CrossRef
20. D. Ruiz and J. Torrens: Residual implications and co-implications from idempotent uninorms. Kybernetika 40 (2004), 21-38.   CrossRef
21. F. Suárez García and P. Gil Álvarez: Two families of fuzzy intergrals. Fuzzy Sets and Systems 18 (1986), 67-81.   CrossRef
22. 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 (1998), 263-274.   CrossRef
23. Z. D. Wang and Y. D. Yu: Pseudo-$t$-norms and implication operators on a complete Brouwerian lattice. Fuzzy Sets and Systems 132 (2002), 113-124.   CrossRef
24. Z. D. Wang: Generating pseudo-$t$-norms and implication operators. Fuzzy Sets and Systems 157 (2006), 398-410.   CrossRef
25. 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.   CrossRef
26. 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.   CrossRef
27. R. R. Yager: Uninorms in fuzzy system modeling. Fuzzy Sets and Systems 122 (2001), 167-175.   CrossRef
28. R. R. Yager: Defending against strategic manipulation in uninorm-based multi-agent decision making. European J. Oper. Res. 141 (2002), 217-232.   CrossRef
29. R. R. Yager and V. Kreinovich: Universal approximation theorem for uninorm-based fuzzy systems modeling. Fuzzy Sets and Systems 140 (2003), 331-339.   CrossRef
30. R. R. Yager and A. Rybalov: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111-120.   CrossRef