Loosely collecting examples of abstractions on finite relational structures:
Figure (abstraction by ‘is related’) shows an abstraction by subsuming directly connected nodes in the original into a single node in the model. An edge in the model indicates a common element of its nodes in the original. For example, the nodes 1, 2 become a single node (1, 2) that is related to e.g. node (2, 3) since they have the element 2 in common.
A practical case of such an abstraction is a structure of walls that is abstracted to rooms with the neighborhood relation, where neighborhood is defined by having a wall in common, as in figure (wall in common). For example, the bath (Ba) has walls in common with the bed room (BR) and the foyer (F).
- The abstraction takes into account only the relationships of the nodes. No further properties are considered.
- The abstraction is total on the original-side and not unique on the model-side.
- The model has more nodes than the original. Is this a contradiction to the reduction property of abstraction? I Don’t think so, since a reduction exists from two connected nodes to a single node. Other opinions?
for an informal definition of reduction in the context of models, see Stachowiak