The \(\vec {r}\) -(quasi)-overlap (grouping) functions can be seen as a generalization of the overlap (grouping) functions from “aggregation” to “pre-aggregation”. The relaxation of monotonicity condition leads to better performance in classification problems. Actually, in addition to monotonicity, the commutativity in the definition is also a restrictive condition, as it requires the objects under discussion to be of equal importance. However, asymmetry is prevalent in practical applications, such as distinguishing foreground from background in image processing. On the other hand, fuzzy implications closely related to overlap (grouping) functions and their extensions are crucial in applications such as fuzzy control. Nevertheless, fuzzy implication operators induced by non-commutative pre-quasi-overlap (grouping) functions remains unexplored. Therefore, this article introduces and analyzes the pre-quasi-pseudo overlap (grouping) functions and related residuated implications. First, we consider the extension of pre-quasi-pseudo overlap (grouping) functions on bounded partially ordered sets and propose the concepts of C-monotone quasi-pseudo overlap (grouping) functions. Then induced conditioned monotonic residuated (co-)implications are defined, and related properties are presented. In particular, the ordinal sum and other construction methods of C-monotone quasi-pseudo overlap (grouping) functions are given. Second, the notions of R-monotone quasi-pseudo overlap (grouping) functions are put forward, and their construction theorems as well as the induced directional monotonic residuated (co-)implications are discussed. Moreover, some results regarding the residual principle are demonstrated. The functions introduced in the paper have a broader range of applications and are expected to perform better in classification and image processing.
