Definition:Subsemigroup

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $T \subseteq S$ such that $\left({T, \circ_{\restriction_T}}\right)$, where $\circ_{\restriction_T}$ is the restriction of $\circ$ to $T$, is a semigroup.

Then $\left({T, \circ_{\restriction_T}}\right)$ is a subsemigroup of $S$.

It is usual, for the sake of simplicity, for the same symbol to be used for both $\circ$ and its restriction.

Thus we refer to $\left({T, \circ}\right)$, and we write:
 * $\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$