Subsemigroup Closure Test

Theorem

To show that an algebraic structure $\struct {T, \circ_T}$ is a subsemigroup of a semigroup $\struct {S, \circ}$, we need to show only that:

$(1): \quad T \subseteq S$

$(2): \quad \circ$ is a closed operation in $T$.

where $\circ_T$ denotes the restriction of $\circ$ to $T$.

From Restriction of Associative Operation is Associative, if $\circ$ is associative on $\struct {S, \circ}$, then it will also be associative on $\struct {T, \circ_T}$.

Thus we do not need to check for associativity in $\struct {T, \circ_T}$, as that has been inherited from its extension $\struct {S, \circ}$.

So, once we have established that $T \subseteq S$, all we need to do is to check for $\circ$ to be a closed operation in $T$.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.1$. Subsets closed to an operation
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): $\S 32.1$ Identity element and inverses