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 $$\circ \restriction_T$$