# Definition:Submagma

## Definition

Let $\struct {S, \circ}$ be a magma.

Let $T \subseteq S$ such that $\struct {T, \circ}$ is a magma.

Then $\struct {T, \circ}$ is a **submagma** of $\struct {S, \circ}$.

This relation can be denoted:

- $\struct {T, \circ} \subseteq \struct {S, \circ}$

### Induced Operation

Let $\struct {S, \circ}$ be a magma.

Let $\struct {T, \circ} \subseteq \struct {S, \circ}$.

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 $\circ$**.

## 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 $\struct {S, \circ}$ is a magma and $\struct {T, \circ} \subseteq \struct {S, \circ}$, then we can say that:

- $\struct {T, \circ}$ is
**contained in $\struct {S, \circ}$ algebraically** - $\struct {S, \circ}$
**algebraically contains**$\struct {T, \circ}$ - $\struct {S, \circ}$ is an
**extension**of $\struct {T, \circ}$ - $\struct {T, \circ}$ is
**embedded**in $\struct {S, \circ}$ - $\struct {T, \circ}$ is
**closed**in $\struct {S, \circ}$ - $\struct {T, \circ}$ is
**stable**in $\struct {S, \circ}$.

## Also see

- Results about
**submagmas**can be found here.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.1$. Subsets closed to an operation - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.5$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Operations