Right Cancellable Elements of Semigroup form Subsemigroup

Theorem

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

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

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

Proof

Let $C_\rho$ be the set of right cancellable elements of $\struct {S, \circ}$:

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

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

Then:

\(\ds a \circ \paren {x \circ y}\)

\(=\)

\(\ds b \circ \paren {x \circ y}\)

\(\ds \leadsto \ \ \)

\(\ds \paren {a \circ x} \circ y\)

\(=\)

\(\ds \paren {b \circ x} \circ y\)

by associativity of $\circ$

\(\ds \leadsto \ \ \)

\(\ds a \circ x\)

\(=\)

\(\ds b \circ x\)

as $y \in C_\rho$

\(\ds \leadsto \ \ \)

\(\ds a\)

\(=\)

\(\ds b\)

as $x \in C_\rho$

\(\ds \leadsto \ \ \)

\(\ds x \circ y\)

\(\in\)

\(\ds C_\rho\)

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

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

$\blacksquare$