## Reflections on Abstractions: Concepts vs Modules for Classification

Abstraction Awareness is about deeper understanding of abstraction, a concept so basic to human thinking. Subsequently abstraction is discussed by the means of basic Graph Theory and Formal Concept Analysis.

Recently we looked at how the mathematical notions of Formal Concept (in property structures) and Graph Module (in relational structures) fit with the abstractional principle of Classification. So, now we take a closer look to see how they compare to each other.

If we think of a graph in terms of its adjacency matrix, the graph in the figure (modules diagram) may look like in (modules matrix). As we see in the upper right triangle, the nodes of each module, i.e. their x-marks, appear the same from the outside (i.e. for other nodes). It doesn’t matter what the boxes ‘I’ to ‘IV’ look like on the inside. This is why Modularisation provides a tree structure of modules with sub modules etc.

A Conceptual Context can be thought of as bi-partite graph. Thus, the lattice in (concepts diagram) would look like (concepts matrix), with 4, 5, 6 serving as properties, and the boxes ‘I’ and ‘II’ empty.

Modularisation wouldn’t take us any further here, since the nodes look all different from the ‘outside’ (i.e. have different combinations of properties). However, conceptualisation does help, since it allows multiple generalisation. On the other hand, conceptualisation puts restrictions on what happens inside ‘I’ and ‘II’, since the ‘wrong’ arcs here, can lead to incompatibilities with the concept lattice, i.e. the view from outside the box.

Thus, classifying relational structures by modules and property structures by formal concepts seems to obey similar, but in detail different principles. Something that deserves a closer look.

So long
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