On triangular norms and uninorms definable in L Pi 1/2
In this paper, we investigate the definability of classes of t-norms and uninorms in the logic L Pi 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 construction methods 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. (c) 2007 Elsevier Inc. All rights reserved.
left-continuous t-norms decidability fuzzy logics semialgebraic sets left-continuous uninorms complexity