Definition:Submagma

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

Let $T \subseteq S$ such that $\left({T, \circ}\right)$ is a magma.

Then $\left({T, \circ}\right)$ is a submagma of $\left({S, \circ}\right)$, and can be denoted:
 * $\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$

Also known as
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.

Some sources, deliberately limiting the quantity of mathematical jargon in their expositions, use neither the term magma nor groupoid.

Under such a limitation, 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 $\left({S, \circ}\right)$
 * $\left({T, \circ}\right)$ is closed in $\left({S, \circ}\right)$
 * $\left({T, \circ}\right)$ is stable in $\left({S, \circ}\right)$.

Also see

 * Definition:Extension of Operation
 * Definition:Restriction of Operation


 * Magma is Submagma of Itself


 * Empty Set is Submagma of Magma


 * Subset not necessarily Submagma

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.