# Relation Type

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A Relation Type is a type that can be associated with a relation instance.

## 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
• $\displaystyle{ T }$ is the set of types ; we will further assume that $\displaystyle{ T }$ 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.
• $\displaystyle{ B }$ 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 $\displaystyle{ M }$ with respect to a canon is a tuple (TCM, TRM, IM), where
• TCM is the set of concept types for the domain $\displaystyle{ M }$ 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, $\displaystyle{ s }$ and t, s is said to have a projection into $\displaystyle{ t }$ if and only if there is a morphism hR : $\displaystyle{ R }$$\displaystyle{ R }$, such that: ∀r$\displaystyle{ R }$ and ∀r$\displaystyle{ R }$, hR(r) = r only if type(r) ≥ type'(r) $\displaystyle{ R }$ is the set of relations, and type : $\displaystyle{ R }$$\displaystyle{ T }$ indicates the type of a relation.