In this paper, we introduce a novel index of transitivity for fuzzy relations based on fuzzy equality. The proposed index evaluates the supremal uniform fuzzy-equality proximity between a given relation and the family of exactly transitive relations induced by a continuous triangular norm. We establish several fundamental properties of the proposed index, including faithfulness, stability with respect to fuzzy equality, invariance under inverse relations, monotonicity under restriction, and bounds derived from transitive closures. Furthermore, we prove that the proposed index is attainable on finite universes. A detailed comparison with the traditional degree of transitivity demonstrates that the new index always provides an evaluation no smaller than the classical measure and can distinguish different aspects of approximate transitivity. Finally, we characterize the special case in which the two indices coincide for all fuzzy relations and show that this occurs precisely for the Gödel triangular norm.

In recent years, vision transformers have demonstrated remarkable superiority to CNNs in single image super-resolution (SISR), yet their heavy computational and storage costs hinder practical deployment. In this work, we revisit CNN-based residual learning for SISR and identify a specific yet overlooked problem: the residual signal is often biased by mapping errors during model training, and this bias can propagate and accumulate through layers. To this end, we revisit the commonly used residual learning and feature fusion in SISR and propose an adaptive residual correction network (AdaRCN) in this work. First, we introduce a residual correction mechanism that adaptively compensates for the bias in the residual, which is utilized to ease error accumulation and improve mapping accuracy. On the other hand, we generalize the standard identity shortcut to a weighted channel concatenation followed by a 1 × 1 convolution, which is a more versatile strategy for adaptive feature fusion. Our AdaRCN is built entirely upon a naive CNN without complex architecture design and training strategies, thus ensuring efficient inference and parallelization. Extensive experiments verify the benefits and effectiveness of residual correction and adaptive feature fusion in improving the representational capability of our model, enabling it to achieve impressive performance comparable to advanced SISR models with moderate overhead.

The formal ordinal sums of t-norms on the bounded lattice is defined by Saminger, but the formal ordinal sum need not be a t-norm. Hence, she provides a necessary and sufficient condition such that the formal ordinal sum is a t-norm indeed. In this paper, we first give several conditions that are equivalent to Saminger’s condition. Secondly, we prove that there is a largest sublattice containing all the pairwise non-overlapped subintervals such that the formal ordinal sum is a t-norm on this sublattice when the bounded lattice is modular.