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 as alternatives to the usual ones between crisp and fuzzy subsets. In particular, the concept of “the non-trivial content of the empty set ∅” is analyzed. This proposed model is based on the interconnection of two consolidated mathematical concepts in the specialized literature: One of them, (which belongs to the field of image processing using Mathematical Morphology techniques), is that of activity ordering and that we use here in the general context of lattices (L, ≤) and in particular in that of Boolean Algebras. The other consists of a version in distributive lattices (L, ≤) of the symmetric difference operator Δ, a classic concept in Set Theory. The utility of the model is illustrated in the following fields: analysis of risk maps, (areas of avalanches, risk of fires, landslides, earthquakes, ...), as well as maps with contour lines: (isochrons, isotherms, salinity , rainfall, intensity of earthquakes, ...). 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 justified 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).