Left and Right Operation are Closed for All Subsets

Theorem

Let $S$ be a set.

Let:

$\leftarrow$ be the left operation on $S$

$\rightarrow$ be the right operation on $S$.

That is:

$\forall x, y \in S: x \leftarrow y = x$

$\forall x, y \in S: x \rightarrow y = y$

Let $\powerset S$ be the power set of $S$.

Then for all $T \in \powerset S$, both $\leftarrow$ and $\rightarrow$ are closed on $T$.

That is, for all $T \in \powerset S$:

Left Operation is Closed for All Subsets

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

Right Operation is Closed for All Subsets

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.1$