Definition:Subsemigroup

A subsemigroup $$\left({T, \circ}\right)$$ of a semigroup $$\left({S, \circ}\right)$$ is a semigroup $$\left({T, \circ}\right)$$ such that $$T \subseteq S$$. We write $$\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$$.

To show that $$\left({T, \circ}\right)$$ is a subsemigroup of $$\left({S, \circ}\right)$$, we need to show only that $$T \subseteq S$$ and that $$\left({T, \circ}\right)$$ is a groupoid, i.e. that $$\left({T, \circ}\right)$$ is closed.

This follows from the definition of restriction, where it is shown that if $$\circ$$ is associative on $$\left({S, \circ}\right)$$, then it will also be associative on $$\left({T, \circ}\right)$$.

Thus we do not need to check for associativity in $$\left({T, \circ}\right)$$ - that has been inherited from its extension $$\left({S, \circ}\right)$$.

All we need to do is to check for $$\left({T, \circ}\right)$$ to be closed.