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.

最近，Paiva等人。在晶格的上下文中概括了重叠函数的概念，并引入了一个较弱的定义，即准重叠，该定义源自去除连续性条件。在本文中，我们介绍了与格上的拟重叠函数有关的剩余含义的概念，并证明了一些相关的性质。我们还表明，根据格上的Scott拓扑，满足残差原理的准重叠函数的类别与连续函数的类别相同。Scott连续性和密集有序的球状体的概念用于归纳格上剩余的拟重叠函数的分类定理。共轭准重叠也被考虑。