# Article

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Keywords:
uninorms; continuity; $t$-norms; $t$-conorms; ordinal sum of semigroups
Summary:
Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation $U$ in the unit interval with the neutral element $e\in [0,1]$. If operation $U$ is continuous, then $e=0$ or $e=1$. So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element $e\in (0,1)$, which is continuous in the open unit square may be given in $[0,1)^2$ or $(0,1]^2$ as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7].
References:
[1] Clifford A. H.: Naturally totally ordered commutative semigroups. Amer. J. Math. 76 (1954), 631–646 MR 0062118
[2] Climescu A. C.: Sur l’équation fonctionelle de l’associativité. Bull. Ecole Polytechn. 1 (1946), 1–16
[3] Czogała E., Drewniak J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets and Systems 12 (1984), 249–269 MR 0740097 | Zbl 0555.94027
[4] Dombi J.: Basic concepts for a theory of evaluation: The aggregative operators. European J. Oper. Res. 10 (1982), 282–293 MR 0665480
[5] Drewniak J., Drygaś P.: Ordered semigroups in constructions of uninorms and nullnorms. In: Issues in Soft Computing Theory and Applications (P. Grzegorzewski, M. Krawczak, and S. Zadrożny, eds.), EXIT, Warszawa 2005, pp. 147–158
[6] Fodor J., Yager, R., Rybalov A.: Structure of uninorms. Internat. J. Uncertain. Fuzziness Knowledge–Based Systems 5 (1997), 411–427 MR 1471619 | Zbl 1232.03015
[7] Hu S.-K., Li Z.-F.: The structure of continuous uninorms. Fuzzy Sets and Systems 124 (2001), 43–52 MR 1859776 | Zbl 1132.03349
[8] Jenei S.: A note on the ordinal sum theorem and its consequence for the construction of triangular norm. Fuzzy Sets and Systems 126 (2002), 199–205 MR 1884686
[9] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[10] Li Y.-M., Shi Z.-K.: Remarks on uninorm aggregation operators. Fuzzy Sets and Systems 114 (2000), 377–380 MR 1775275 | Zbl 0962.03052
[11] Mas M., Monserrat, M., Torrens J.: On left and right uninorms. Internat. J. Uncertain. Fuzziness Knowledge–Based Systems 9 (2001), 491–507 MR 1852342 | Zbl 1045.03029
[12] Sander W.: Associative aggregation operators. In: Aggregation Operators (T. Calvo, G. Mayor, and R. Mesiar, eds), Physica–Verlag, Heidelberg 2002, pp. 124–158 MR 1936386 | Zbl 1025.03054
[13] Yager R., Rybalov A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111–120 MR 1389951 | Zbl 0871.04007

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