ON THE PROPERTIES OF A GENERALIZED CLASS OF T-NORMS IN INTERVAL-VALUED FUZZY LOGICS

Since it does not generate any MTL-algebra (prelinear residuated lattice), the lattice $\mathcal{L}^I$ of closed subintervals of [0, 1] falls outside the mainstream of research on formal fuzzy logics. However, due to the intimate connection between logical connectives on $\mathcal{L}^I$ and those on [0, 1], many relevant logical properties can still be maintained, sometimes in a slightly weaker form. In this paper, we focus on a broad class of parametrized t-norms on $\mathcal{L}^I$. We derive their corresponding residual implicators, and examine commonly imposed logical properties. Importantly, we formally establish one-to-one correspondences between ∨-definability (respectively, weak divisibility) for t-norms of this class and strong ∨-definability (resp., divisibility) for their counterparts on [0, 1].