# Relation Type

## References

### 2008

• (Corbett, 2008) ⇒ Dan R. Corbett. (2008). “Graph-based Representation and Reasoning for Ontologies.” In: Studies in Computational Intelligence, Springer. [http://dx.doi.org/10.1007/978-3-540-78293-3 10.1007/978-3-540-78293-3 doi:[http://dx.doi.org/10.1007/978-3-540-78293-3 10.1007/978-3-540-78293-3)
• QUOTE: A canon is a tuple (T, I, =, ::, B), where
• [math]T[/math] is the set of types ; we will further assume that [math]T[/math] contains two disjunctive subsets TC and TR containing types for concepts and relations.
• \$I\$ is the set of individuals.
• ≤ ⊆ T×T is the subtype relation. It is assumed to be a lattice (so there are types `T` and ⊥ and operations ∧ and ∨).
•  :: ⊂ I×T is the conformity relation. The conformity relation relates type labels to individual markers; this is essentially the relation which ensures that the typing of the concepts makes sense in the domain, and helps to enforce the type hierarchy.
• [math]B[/math] is the Canonical Basis function (also called s in the Conceptual Graphs literature). This function associates each relation type with the concept types that may be used with that relation; this helps to guarantee well-formed graphs.
• QUOTE: An ontology in a given domain [math]M[/math] with respect to a canon is a tuple (TCM, TRM, IM), where
• TCM is the set of concept types for the domain [math]M[/math] and TRM is the set of relation types for the domain M.
• IM is the set of individuals for the domain M.
• Given two relation types, [math]s[/math] and t, s is said to have a projection into [math]t[/math] if and only if there is a morphism hR : [math]R[/math][math]R[/math], such that: ∀r[math]R[/math] and ∀r[math]R[/math], hR(r) = r only if type(r) ≥ type'(r) [math]R[/math] is the set of relations, and type : [math]R[/math][math]T[/math] indicates the type of a relation.