Orders on Partial Partitions and Maximal Partitioning of Sets

Abstract

The segmentation of a function on a set can be considered as the construction of a maximal partial partition of that set with blocks satisfying some criterion for the function. Several order relations on partial partitions are considered in association with types of operators and criteria involved in the segmentation process. We investigate orders for which this maximality of the segmentation partial partition is preserved in compound segmentation with two successive criteria. Finally we consider valuations on partial partitions, that is, strictly isotone functions with positive real values; this gives an alternative approach where the valuation, not the partial partition, should be maximized.