Left Cancellable Elements of Semigroup form Subsemigroup

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let $C_\lambda$ be the set of left cancellable elements of $\struct {S, \circ}$.

Then $\struct {C_\lambda, \circ}$ is a subsemigroup of $\struct {S, \circ}$.

Proof
Let $C_\lambda$ be the set of left cancellable elements of $\struct {S, \circ}$:


 * $C_\lambda = \set {x \in S: \forall a, b \in S: x \circ a = x \circ b \implies a = b}$

Let $x, y \in C_\lambda$.

Then:

Thus $\struct {C_\lambda, \circ}$ is closed.

Therefore by the Subsemigroup Closure Test $\struct {C_\lambda, \circ}$ is a subsemigroup of $\struct {S, \circ}$.