## On a family of t-norms.(English)Zbl 0739.39006

The authors consider the functional equation (1) $$T(x,y)+| x- y|=T(\hbox{Max}(x,y)$$, $$\hbox{Max}(x,y))$$, where $$T$$ is a $$t$$-norm. (A $$t$$-norm is a binary operation on $$[0,1]$$ that is associative, commutative, non-decreasing in each place, and has identity 1.) The authors prove that if $$T$$ is a continuous $$t$$-norm which satisfies (1), then, for some $$\alpha\in[0,1]$$, $$T$$ is of the form $$T(x,y)=\hbox{Max}(0,x+y-\alpha)$$, whenever $$x,y\in[0,\alpha]$$, and $$=\hbox{Min}(x,y)$$, otherwise.

### MSC:

 39B22 Functional equations for real functions 03E72 Theory of fuzzy sets, etc.

### Keywords:

strong negation; functional equation; continuous $$t$$-norm
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### References:

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