On \vec-(Quasi-)Overlap Functions

Overlap functions, as one kind of particular binary aggregation functions, have been continuously studied in the literature for their vast preponderance in some real applications. Meanwhile, after Lucca et al. introduced the notion of preaggregation functions recently, the investigation related to such "aggregation" functions becomes an interesting, and natural research area. In this article, we continue to focus on this topic for overlap functions. First, we introduce the concept of \vec{r}-(quasi-)overlap functions by replacing the monotonicity of (quasi-)overlap functions with directional monotonicity, and the notions of 0-\vec{r}-(quasi-)overlap functions, 1-\vec{r}-(quasi-)overlap function, and 0,1-\vec{r}-(quasi-)overlap functions associated with them. And then, we propose some vital properties of \vec{r}-(quasi-)overlap functions. Finally, we show several construction methods of \vec{r}-(quasi-)overlap functions, 0-\vec{r}-(quasi-)overlap functions, 1-\vec{r}-(quasi-)overlap function, and 0,1-\vec{r}-(quasi-)overlap functions.