Conditioned Monotonicity for Generalized Pre-Aggregations and Aggregations

Abstract

The concept of pre-aggregation function defined in [0,1]n has been recently extended to that of generalized pre-aggregation function in the framework of a totally ordered set $\mathcal{T}$ with maximum and minimum value. To do so, the concept of monotonicity is transformed in that of conditioned monotonicity based on the chains in ${\mathcal{T}^n}$, generalizing the idea of directional monotonicity. In the present paper we explore the concept of conditioned monotonicity considering some specific conditioning structures (covers, partitions and projections). On this basis we consider some situations where conditioned monotonicity ensures monotonicity. Finally we use these definitions and properties to define some pre-aggregation and aggregation functions that are applied to image preprocessing problems.