# What does canonical extension mean?

Density-reachability is a canonical extension of direct density-reachability. What is meant by canonical extension?

Source: A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise https://www.aaai.org/Papers/KDD/1996/KDD96-037.pdf

• Could you give the source of the quote (with hyperlink if possible)? – Peter Taylor May 1 '18 at 11:15
• @PeterTaylor I added the title and a direct link to the paper where it was used – Vincent May 2 '18 at 11:19

The term 'extension' could generally be interpreted in mathematical sense1 as an enlargement of the domain of a pre-existing function.

In your case, assume Density-reachability could be described as a process $d\colon A \rightarrow B$ mapping inputs from domain $A$ to output in domain $B$. An enlargement of domain could be viewed as another map $e\colon A \rightarrow A'$ which maps where the old inputs in $A$ could be thought of as special cases in subset of $A'$ that is the image of map $e$.

An extension then could be described as any map $x\colon A' \rightarrow B$ which statisfy $$x \circ e = d$$ There could be many such $x$, so picking extension is a choice. The above equation basically expresses that

If I pick $a \in A$ from the old domain, translate it to the new domain as $e(a)$, then use this new tool $x$ which could be thought of as an extension of old tool $d$ along the translation $e$, then the final result must agree with just using the old tool $d$ to begin with.

or $$x \circ e(a) = d(a)\qquad; \forall a \in A$$

The term 'canonical' simply implies that the choice of extension is the obvious, natural one of the situation.

1 At least from the perspective of category theory.