Definition:Submagma

Definition
If:
 * $$\left({S, \circ}\right)$$ is a groupoid;
 * $$T \subseteq S$$;
 * $$\left({T, \circ}\right)$$ is a groupoid;

then $$\left({T, \circ}\right)$$ is a subgroupoid of $$\left({S, \circ}\right)$$, and we can write $$\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$$.

Note the following.

Suppose $$\left({S, \circ}\right)$$ is a groupoid.

Suppose $$T \subseteq S$$.

Suppose $$\exists s, t \in T: s \circ t \notin T$$, although of course $$s \circ t \in S$$.

Then $$\left({T, \circ}\right)$$ is not closed, and it is not true to write $$\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$$.

This is because $$\left({T, \circ}\right)$$ is not actually a groupoid itself, through dint of it not being closed.

If $$\left({S, \circ}\right)$$ is a groupoid and $$\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$$, then we can say that:


 * $$\left({T, \circ}\right)$$ is contained in $$\left({S, \circ}\right)$$ algebraically;
 * $$\left({S, \circ}\right)$$ algebraically contains $$\left({T, \circ}\right)$$;
 * $$\left({S, \circ}\right)$$ is an extension of $$\left({T, \circ}\right)$$;
 * $$\left({T, \circ}\right)$$ is embedded in $$\left({S, \circ}\right)$$.

Induced Operation
Let $$\left({S, \circ}\right)$$ be a groupoid and let $$\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$$.

Then the restriction of $\circ$ to $T$, namely $$\circ \restriction_T$$, is called the binary operation induced on $$T$$ by $$\circ$$.

Note that this definition applies only if $$\left({T, \circ}\right)$$ is closed, by which virtue it is a subgroupoid of $$\left({S, \circ}\right)$$.

Obvious subgroupoids
If $$\left({S, \circ}\right)$$ is a groupoid, then $$\left({S, \circ}\right)$$ is always a subgroupoid of $$\left({S, \circ}\right)$$. That is, all groupoids are subgroupoids of themselves.

If $$\left({S, \circ}\right)$$ is a groupoid, then $$\left({\varnothing, \circ}\right)$$ is always a subgroupoid of $$\left({S, \circ}\right)$$. That is, the empty set is always a subgroupoid of any groupoid:

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