Definition:Operation Induced by Restriction

Definition

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 $\circ$.

Note that this definition applies only if $\struct {T, \circ}$ is closed, by which virtue it is a submagma of $\struct {S, \circ}$.

Also known as

The notation $\circ_T$ is also found for $\circ {\restriction_T}$.

Also see

• Definition:Restriction of Operation

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets