Reflections on Abstractions: Joining Classification by Relationships and Properties

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.

What’s happened so far

What happens hereThis is just a little add-on to the former posting Concepts vs Modules for Classificationwhere remained open, if and how the relationships inside the sub-matrix ‘I’ in the figure (concept matrix) can be compatible to the concept lattice, defined by the upper right sub-matrix.

The nice case

concept lattice relationshipThe four cases of ‘compatible’ relationships are shown in figure (concepts and relations) up to isomorphy. The corresponding matrix views are shown in figures (a) to (d). For example, relationship (a) from concept 1 to 2, element (1,2) in the matrix, implies also the matrix element (2,2) since concept 1 is a generalisation of 2.
relation concept cases

The not so nice case

fitted latticeHowever, sometimes the concept lattice may need some amendment. Notice, the above are the ‘compatible’ cases, where the concept lattice is in alignment with the relationships ‘seamlessly’.  In other cases, like in figure (concepts and relations matrix) the concept lattice might need to be fitted correspondingly like (fitted lattice).

Altogether …

we’ve seen some basic cases how property structures and relational structures can fit together, more or less ‘seamlessly’.

So long
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About modelpractice

Modeling Theory and Abstraction Awareness in the gap btw rigour of science and relevance to engineering.
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One Response to Reflections on Abstractions: Joining Classification by Relationships and Properties

  1. There is certainly a great deal to learn about this issue.
    I like all of the points you made.

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