Generalized R′-Implications: A Hyper Class of R- and R′-Implications

In this paper, generalized ${\mathrm{R}}^{\prime }$ -implications are introduced and studied. Although they are not always fuzzy implications, the sufficient and necessary condition such that a generalized ${\mathrm{R}}^{\prime }$ -operation is a fuzzy implication is discovered and proved. The satisfaction or violation of some basic properties of fuzzy implications, such as the left neutrality property, the identity principle, and the left ordering property, is also investigated and presented. The intersections among the classes of generalized ${\mathrm{R}}^{\prime }$ -implications (respectively, operations), R-implications, and ${\mathrm{R}}^{\prime }$ -implications (respectively, operations) are also studied. It is demonstrated that the generalized ${\mathrm{R}}^{\prime }$ -implications class is a hyper class of the known R- and ${\mathrm{R}}^{\prime }$ -implications classes. This research also completes the investigation into ${\mathrm{R}}^{\prime }$ -operations and improves the outcomes of the intersection of R- and ${\mathrm{R}}^{\prime }$ -implications.