# Definition:Subsemigroup

## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $T \subseteq S$ such that $\struct {T, \circ {\restriction_T} }$, where $\circ {\restriction_T}$ is the restriction of $\circ$ to $T$, is a semigroup.

Then $\struct {T, \circ {\restriction_T} }$ 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 $\struct {T, \circ}$, and we write:

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

## Examples

### Matrices of the Form $\begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix}$

Let $\struct {S, \times}$ be the semigroup formed by the set of order $2$ square matrices over the real numbers $R$ under (conventional) matrix multiplication.

Let $T$ be the subset of $S$ consisting of the matrices of the form $\begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix}$ for $x \in \R$.

Then $\struct {T, \times}$ is a subsemigroup of $\struct {S, \times}$.

### Operation Defined as $x + y - x y$ on $\Z_{\le 1}$

Let $\struct {\Z, \circ}$ be the semigroup where $\circ: \Z \times \Z$ is the operation defined on the integers $\Z$ as:

$\forall x, y \in \Z: x \circ y := x + y - x y$

Let $T$ be the set $\set {x \in \Z: x \le 1}$.

Then $\struct {T, \circ}$ is a subsemigroup of $\struct {\Z, \circ}$.