Definition:Extension of Operation

Definition
Let $\left({S, \circ}\right)$ be a groupoid.

Let $\left({T, \circ \restriction_T}\right)$ be a subgroupoid of $\left({S, \circ}\right)$, where $\circ \restriction_T$ denotes the restriction of $\circ$ to $T$.

Then:
 * $\left({S, \circ}\right)$ is an extension of $\left({T, \circ \restriction_T}\right)$

or
 * $\left({S, \circ}\right)$ extends $\left({T, \circ \restriction_T}\right)$

We can use the term directly to the operation itself and say:
 * $\circ$ is an extension of $\circ \restriction_T$

or:
 * $\circ$ extends $\circ \restriction_T$