On modelling the concept: "The Contents of the Empty Set" using the Activity Ordering family (⊑^w)w∈L in a distributive lattice (L, ≤). An interpretation of these ordering relations ⊑^w as alternative inclusions and their associated inf-operators ⨅^w

A Mathematical Model is presented to define new inclusions, intersections and unions of crisp and fuzzy subsets. In particular, the concept of “the non-trivial content of the empty set ∅” is analysed. This proposed model is based on the combination of two consolidated mathematical concepts in specialised literature: One of them, (which belongs to the field of image processing using Mathematical Morphology techniques), is the activity ordering whicht we use here in the general context of lattices (L, ≤) and in particular in that of Boolean Algebras. The other is a version in distributive lattices (L, ≤) of the symmetric difference operator Δ, a classical concept in Set Theory. The usefulness of the model is illustrated in the following areas: analysis of risk maps, (avalanche areas, fire risk, landslides, earthquakes, ...), as well as maps with contour lines: (isochrons, isotherms, salinity , rainfall, earthquake intensity, ...). Also in data pre-processing for “Data Mining” tasks and in “Data Analysis with Uncertainty”. A special section is dedicated to the application of the model in Digital Image Processing using Mathematical Morphology techniques. Finally, it is argued that the model can be useful in other fields such as Analysis of Formal Concepts, Probability and in theoretical contexts such as Topology. (Remark. Correction of detected errors).