nilpotent elements.) If is a left-continuous t-norm, then its residuum R is given as follows: where Ri is the residuum of Ti, for each i in I. Ordinal sums of continuous t-norms The ordinal sum of a family of continuous t-norms is a continuous t-norm. By the Mostert–Shields theorem, every continuous t-norm is expressible as the ordinal sum of Archimedean continuous t-norms. Since the latter are either nilpotent (and then isomorphic to the Łukasiewicz t-norm) or strict (then isomorphic to the product t-norm), each continuous t-norm is isomorphic to the ordinal sum of Łukasiewicz and product t-norms. Important examples of ordinal sums of continuous t-norms are the following ones: Dubois–Prade t-norms, introduced by Didier Dubois and Henri Prade in the early 1980s, are the ordinal sums of the product t-norm on [0, p] for a parameter p in [0, 1] and the (default) minimum t-norm on the rest of the unit interval. The family of Dubois–Prade t-norms is decreasing and continuous with respect to p.. Mayor–Torrens t-norms, introduced by Gaspar Mayor and Joan Torrens in the early 1990s, are the ordinal sums of the Łukasiewicz t-norm on [0, p] for a parameter p in [0, 1] and the (default) minimum t-norm on the rest of the unit interval. The family of Mayor–Torrens t-norms is decreasing and continuous with respect to p.. 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: [0, 1] → [0, 1] the function that assigns 1 − x to x and t = 0.5. Let T1 be the linear transformation of T into [t, 1] and Then the function is a left-continuous t-norm, called the rotation of the t-norm T. Geometrically, the construction can be described as first shrinking the t-norm T to the interval [0.5, 1] 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 [0, 1], 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