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:

$\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$

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 (or stable) in $\left({S, \circ}\right)$.

Induced Operation

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

Let $\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$.

That is, let $T$ be a subset of $S$ such that $\circ$ is closed in $T$.

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

Examples

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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \neg \ \exists x, y\)	\(\in\)	\(\displaystyle \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \neg \ \exists x, y\)	\(\in\)	\(\displaystyle \varnothing:\)	\(\displaystyle \)	\(\displaystyle x \circ y \notin \varnothing\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \forall x, y\)	\(\in\)	\(\displaystyle \varnothing:\)	\(\displaystyle \)	\(\displaystyle x \circ y \in \varnothing\)	\(\displaystyle \)

Subset which is not a 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.

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.

Also see

• Extension of an Operation
• Restriction of an Operation

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.1$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 8$