## 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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