This post is part of a series on core relations for structuring representations or models of knowledge.
The aggregation relation is the relation between a whole and its parts. It is the topic of mereology. It is considered to be one of basic abstraction principles [Taivalsaari:1996] in computer science. “Aggregation relation” is synonym to “Decomposition relation”, and “PartOf relation”. They are all names of relations between parts and a whole.
The aggregation relation satisfies the following rules [Borgo+:2009]:
- Reflexivity: An X is always part of itself.
- Transitivity: If X is part of Y, and Y is part of Z, then X is part of Z.
- Antisymmetry: If X is part of Y, and Y is part of X, then X and Y are equal.
- Extensionality: If X is not part of Y, then there must exist some Z such that Z is part of X and Z is not part of Y.
- Dissectivity: If Y has a property P and X is part of Y, then X has property P.
- Additivity: If X has property P and Y has property P, and Z has no parts other than X and Y, then Z has property P.
There are other, alternative sets of rules that you may want aggregation to satisfy; I like the above, because I agree with all but the Dissectivity rule: It is not clear to me how emergent properties (those that a whole has, and none of its parts does) fit in the above, since they seem to violate Dissectivity.