Rotations
The construction of t-norms by rotation was introduced by Sándor Jenei (2000). It is based on the following theorem:
- Let T be a left-continuous t-norm without zero divisors, N: → the function that assigns 1 − x to x and t = 0.5. Let T1 be the linear transformation of T into and Then the function
- is a left-continuous t-norm, called the rotation of the t-norm T.
![T_{\mathrm{rot}} = \begin{cases} T_1(x, y) & \mbox{if } x, y \in (t, 1] \\ N(R_{T_1}(x, N(y))) & \mbox{if } x \in (t, 1] \mbox{ and } y \in \\ N(R_{T_1}(y, N(x))) & \mbox{if } x \in \mbox{ and } y \in (t, 1] \\ 0 & \mbox{if } x, y \in
\end{cases}](http://upload.wikimedia.org/math/3/3/a/33ae5148ca39d31bfcf88134ce3b9a92.png)
Geometrically, the construction can be described as first shrinking the t-norm T to the interval and then rotating it by the angle 2π/3 in both directions around the line connecting the points (0, 0, 1) and (1, 1, 0).
The theorem can be generalized by taking for N any strong negation, that is, an involutive strictly decreasing continuous function on, and for t taking the unique fixed point of N.
The resulting t-norm enjoys the following rotation invariance property with respect to N:
- T(x, y) ≤ z if and only if T(y, N(z)) ≤ N(x) for all x, y, z in .
The negation induced by Trot is the function N, that is, N(x) = Rrot(x, 0) for all x, where Rrot is the residuum of Trot.
Read more about this topic: Construction Of T-norms