A closure of a relation
Rel is the most minimal relation with respect to some property
Prop that contains the source relation
Thinking computationally, the closure operation takes a relation and adds the least number of relations
needed so that the resulting relation fulfills the property
If we think about the relationship between any closure to the source relation
Rel, one might
note the following rule holds:
Closure (with respect to any property) contains the source relation as a subset.
Closure is more information rich than the source relation.
There are also two other neat properties closures have:
Applying a closure on a relation which already has closure results in the same relation.
Repeated application of closure produces no new information.
Closure operation respects the underlying containment relations or that “the diagram commutes”.
Information gained by applying a transitive closure on a subset SubRel C would be a subset of the one gained by applying it on its superset Rel C.
Note that since the closure property is extensive, once a closure relation is applied to a relation, one might need to apply the reverse process called kernel operation to make it acquire arbitrary desired properties. Otherwise, if closure operation is the only one we have in our repertoire, we would be left with operations that only increase the information but not reduce it. This aspect will be explored in a later outing in this blog post.
Reflexive and Reflexive Transitive Closure
Symmetric and Symmetric Transitive Closure
Leibniz’ Theory of Concepts
Leibniz on Intension and Extension
The Algebra of Logic Tradition
Origins of the Calculus of Relations
The Origins of Relation Algebras in the Development and Axiomatization of the Calculus of Relations
Normal Forms and Reduction for Theories of Binary Relations
First Steps in Pointfree Dependency Theory