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:
 * $\struct {T, \circ} \subseteq \struct {S, \circ}$

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

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


 * Magma is Submagma of Itself


 * Empty Set is Submagma of Magma


 * Subset not necessarily Submagma