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Structure of left-continuous triangular norms with strong induced negations. III: Construction and decomposition. (English) Zbl 1050.03505

Summary: This paper is the continuation of Parts I and II [J. Appl. Non-Class. Log. 10, 83–92 (2000; Zbl 1033.03512); ibid. 11, 351–366 (2001; Zbl 1037.03508)], where the rotation construction and the rotation-annihilation construction have been presented, respectively. Both constructions produce left-continuous (but not continuous) triangular norms with strong induced negations. Here, we show that these two constructions allow us to define indecomposability and that each decomposable left-continuous triangular norm with strong induced negation can be derived as result of one of the above constructions. In this way two general decomposition theorems arise. In addition, the decomposition of left-continuous triangular norms with strong induced negations can be defined in a unique way, namely, when the decomposition is done by the minimal decomposition point.

MSC:

03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
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