*Loosely collecting examples of abstractions on finite relational structures:*

### Relational World

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.

### Real World

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).

### Remarks

- 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?

So long

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PS

for an informal definition of reduction in the context of models, see Stachowiak

Good example! But I still don’t like the term “reduction” (or such the simplification, abbreviation) for modeling, since the term abstraction (maybe, selection) is enough and stranger than “reduction”. 😉

Thanks! So is ‘removed’ the right word in your terms? In this terms, would it be right to say the nodes from the original are removed by subsuming them into a new node in the model?

So long

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No, perhaps ‘removed’ is worse.

I think, ‘to make a model’ is not ‘to process or modify the original entity’ but ‘to make a new entity that has some properties corresponding to the properties of the original’.

In the first picture, each node of the models is corresponding to one relation (edge) of the original, and each edge of the model is corresponding to one edges’ relation (a pair of edges have a common node) of the original. The interpretation like “subsuming two nodes of original into one node of model” is IMO a bad expression.

yes, good point to mention. systematisation of abstractions is sometimes tricky. abstraction ‘nodes to nodes by edges’ can sometimes lead to the same result as abstracting ‘edges to nodes’.

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