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 $\struct {\Z, \circ}$.
Also see
- Results about subsemigroups can be found here.
Generalizations
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 32$ Identity element and inverses