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:


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

Also see

 * Definition:Subgroup
 * Definition:Submonoid

Generalizations

 * Definition:Algebraic Substructure