## Different types of continuity of triangular norms revisited.(English)Zbl 1081.26024

The authors study many continuity conditions for t-norms $$T$$, namely (in the order of decreasing generality): – continuity at $$(1,1)$$, – broder continuity (continuity at all points of $$\{1\}\times[0,1]$$), – left continuity, – continuity, – $$k$$-Lipschitz property for $$k>1$$, – $$1$$-Lipschitz property, – $$\| \cdot\| _p$$-stability ($$| T(x_1,y_1)-T(x_2,y_2)| \leq(| x_1-x_2| ^p+| y_1-y_2| ^p)^{1/p}$$), $$p>1$$. Numerous results on these types of continuity are collected, in particular their relations to ordinal sums and additive generators. Particular attention is paid to the correspondence with the properties of diagonal sections, i.e., the functions $$x\mapsto T(x,x)$$. Also Schur concavity, $T(x,y)\leq T(\lambda x+(1-\lambda)y, (1-\lambda)x+\lambda y)$ for all $$\lambda\in[0,1]$$ and its special form for $$\lambda=1/2$$, $T(x,y)\leq T((x+y)/2,(x+y)/2),$ are put into this context. It is a useful and complete survey of contemporary knowledge in this field, accompanied by numerous interesting examples.

### MSC:

 2.6e+51 Fuzzy real analysis 3e+72 Theory of fuzzy sets, etc.
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### References:

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