Aggregation of random elements over bounded lattices

Aggregation functions are widely used to fuse information from different sources in a unique value. In many cases, the aggregated information is related to some experimental measure or random sampling of a population. In this direction, it is reasonable to consider aggregation of random elements. In this paper, the concept of aggregation functions of random elements over bounded lattices, which are measurable functions from a probability space to a bounded lattice, is presented. In particular, starting from a partially ordered set, a measurable space is constructed. Random elements are considered to be measurable functions from a probability space to the measurable space. The concept of aggregation of random elements over bounded lattices is defined by generalizing the monotonicity and the boundary conditions in terms of stochastic orders. Several types, such as the induced, random and degenerated aggregations of random elements over bounded lattices are defined and some coherence properties are studied. Particular examples regarding the aggregation of random variables, random graphs and random semi -positive matrices are provided.