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Left and right semi-uninorms on a complete lattice. (English) Zbl 1286.03098
Summary: 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.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03E72 Theory of fuzzy sets, etc. 06B23 Complete lattices, completions
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##### References:
 [1] Baets, B. De: Coimplicators, the forgotten connectives. Tatra Mountains Math. Publ. 12 (1997), 229-240. · Zbl 0954.03029 [2] Baets, B. De: Idempotent uninorms. European J. Oper. Res. 118 (1999), 631-642. · Zbl 1178.03070 [3] Baets, B. De, Fodor, J.: Van Melle’s combining function in MYCIN is a representable uninorm: an alternative proof. Fuzzy Sets and Systems 104 (1999), 133-136. · Zbl 0928.03060 [4] Bassan, B., Spizzichino, F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93 (2005), 313-339. · Zbl 1070.60015 [5] Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publishers, Providence 1967. · Zbl 0537.06001 [6] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra. Springer-Verlag, New York 1981. · Zbl 0478.08001 [7] Cooman, G. De, Kerre, E. E.: Order norms on bounded partially ordered sets. J. Fuzzy Math. 2 (1994), 281-310. · Zbl 0814.04005 [8] Durante, F., Klement, E. P., al., R. Mesiar et: Conjunctors and their residual implicators: characterizations and construct methods. Mediterranean J. Math. 4 (2007), 343-356. · Zbl 1139.03014 [9] Fodor, J., Yager, R. R., Rybalov, A.: Structure of uninorms. Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems 5 (1997), 411-427. · Zbl 1232.03015 [10] Gabbay, D., Metcalfe, G.: fuzzy logics based on $$[0,1)$$-continuous uninorms. Arch. Math. Logic 46 (2007), 425-449. · Zbl 1128.03015 [11] Gottwald, S.: A Treatise on Many-Valued Logics. Studies in Logic and Computation Vol. 9, Research Studies Press, Baldock 2001. · Zbl 1048.03002 [12] Jenei, S.: A characterization theorem on the rotation construction for triangular norms. Fuzzy Sets and Systems 136 (2003), 283-289. · Zbl 1020.03021 [13] Jenei, S.: How to construct left-continuous triangular norms-state of the art. Fuzzy Sets and Systems 143 (2004), 27-45. · Zbl 1040.03021 [14] Jenei, S., Montagna, F.: A general method for constructing left-continuous $$t$$-norms. Fuzzy Sets and Systems 136 (2003), 263-282. · Zbl 1020.03020 [15] Liu, H. W.: Semi-uninorm and implications on a complete lattice. Fuzzy Sets and Systems 191 (2012), 72-82. · Zbl 1244.03091 [16] Ma, Z., Wu, W. M.: Logical operators on complete lattices. Inform. Sci. 55 (1991), 77-97. · Zbl 0741.03010 [17] Mas, M., Monserrat, M., Torrens, J.: On left and right uninorms. Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems 9 (2001), 491-507. · Zbl 1045.03029 [18] Mas, M., Monserrat, M., Torrens, J.: On left and right uninorms on a finite chain. Fuzzy Sets and Systems 146 (2004), 3-17. · Zbl 1045.03029 [19] Mas, M., Monserrat, M., Torrens, J.: Two types of implications derived from uninorms. Fuzzy Sets and Systems 158 (2007), 2612-2626. · Zbl 1125.03018 [20] Ruiz, D., Torrens, J.: Residual implications and co-implications from idempotent uninorms. Kybernetika 40 (2004), 21-38. · Zbl 1249.94095 [21] García, F. Suárez, Álvarez, P. Gil: Two families of fuzzy intergrals. Fuzzy Sets and Systems 18 (1986), 67-81. · Zbl 0595.28011 [22] Tsadiras, A. K., Margaritis, K. G.: 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. [23] Wang, Z. D., Yu, Y. D.: Pseudo-$$t$$-norms and implication operators on a complete Brouwerian lattice. Fuzzy Sets and Systems 132 (2002), 113-124. · Zbl 1013.03020 [24] Wang, Z. D.: Generating pseudo-$$t$$-norms and implication operators. Fuzzy Sets and Systems 157 (2006), 398-410. · Zbl 1085.03020 [25] Wang, Z. D., Fang, J. X.: Residual operators of left and right uninorms on a complete lattice. Fuzzy Sets and Systems 160 (2009), 22-31. · Zbl 1183.06003 [26] Wang, Z. D., Fang, J. X.: Residual coimplicators of left and right uninorms on a complete lattice. Fuzzy Sets and Systems 160 (2009), 2086-2096. · Zbl 1183.03027 [27] Yager, R. R.: Uninorms in fuzzy system modeling. Fuzzy Sets and Systems 122 (2001), 167-175. · Zbl 0978.93007 [28] Yager, R. R.: Defending against strategic manipulation in uninorm-based multi-agent decision making. European J. Oper. Res. 141 (2002), 217-232. · Zbl 0998.90046 [29] Yager, R. R., Kreinovich, V.: Universal approximation theorem for uninorm-based fuzzy systems modeling. Fuzzy Sets and Systems 140 (2003), 331-339. · Zbl 1040.93043 [30] Yager, R. R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111-120. · Zbl 0871.04007
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