Residuated implications derived from quasi-overlap functions on lattices

Recently, Paiva et al. generalized the notion of overlap functions in the context of lattices and introduced a weaker definition, called quasi-overlap, that originates from the removal of the continuity condition. In this paper, we introduce the concept of residuated implications related to quasi-overlap functions on lattices and prove some related properties. We also show that the class of quasi-overlap functions that fulfill the residuation principle is the same class of continuous functions according to a Scott topology on lattices. Scott continuity and the notion of densely ordered posets are used to generalize a classification theorem for residuated quasi-overlap functions on lattices. Conjugated quasi-overlaps are also considered.