Definition:Submagma

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

then $\left({T, \circ}\right)$ is a submagma 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 magma.

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 magma itself, through dint of it not being closed.

If $\left({S, \circ}\right)$ is a magma 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 magma 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 submagma of $\left({S, \circ}\right)$.

Obvious submagmas
If $\left({S, \circ}\right)$ is a magma, then $\left({S, \circ}\right)$ is always a submagma of $\left({S, \circ}\right)$. That is, all magmas are submagmas of themselves.

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

Alternative names
An older term for this concept is subgroupoid (or sub-gruppoid), from groupoid.

A groupoid is now often understood to be a concept in category theory.

Also see

 * Extension of an Operation
 * Restriction of an Operation