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On triangular norms and uninorms definable in Ł\(\Pi \frac{1}{2}\). (English) Zbl 1189.03032
Summary: We investigate the definability of classes of t-norms and uninorms in the logic Ł\(\Pi \frac{1}{2}\). In particular, we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on \([0, 1)\), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under certain construction methods, such as annihilation, rotation and rotation-annihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally, we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms.

MSC:
03B52 Fuzzy logic; logic of vagueness
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