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 $\circ {\restriction_T}$.