Definition:Submagma

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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

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