Pointwise directional increasingness and geometric interpretation of directionally monotone functions.

The relaxation of monotonicity requirements is a trend in the theory of aggregation functions. In the recent literature, we can find several relaxed forms of monotonicity, such as weak, directional, cone, ordered directional and strengthened directional monotonicity. All these forms of monotonicity are global properties in the sense that they are imposed for all the points in the domain of a function. In this work, we introduce a local notion of monotonicity called pointwise directional monotonicity, or directional monotonicity at a point. Based on this concept, we characterize all the previously defined notions of monotonicity and, in the final part of the paper, we present some geometric aspects of the global weaker forms of monotonicity, stressing their relations and singularities.