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The functional equations of Frank and Alsina for uninorms and nullnorms. (English) Zbl 0977.03026
Motivated by some questions arising in fuzzy preference modelling, the authors consider some generalizations of Frank’s functional equation: $T(x,y) + S(x,y) = x+ y \tag{1}$ and Alsina’s functional equation $S(T(x,y), T(x,N(y)) = x . \tag{2}$ Equations (1) and (2) were solved initially in the case that $$T$$ is a continuous t-norm, $$S$$ is a continuous t-conorm, $$N$$ is a strong negation and $$x,y$$ run in [0,1]. The chief concern in this paper is to study (1) and (2) for more general classes of binary operations, the so-called uninorms and nullnorms. It is interesting that the solution found led back to the already known solutions for the case of t-norms and t-conorms.

##### MSC:
 03E72 Theory of fuzzy sets, etc. 39B22 Functional equations for real functions 03B52 Fuzzy logic; logic of vagueness
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##### References:
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