All Elements of Right Operation are Right Zeroes

Theorem
Let $\struct {S, \to}$ be an algebraic structure in which the operation $\to$ is the right operation.

Then no matter what $S$ is, $\struct {S, \to}$ is a semigroup all of whose elements are right zeroes.

Thus it can be seen that any right zero in a semigroup is not necessarily unique.

Proof
It is established in Element under Right Operation is Left Identity that $\struct {S, \to}$ is a semigroup.

From the definition of right operation:
 * $\forall x, y \in S: x \to y = y$

from which it can immediately be seen that all elements of $S$ are indeed right zeroes.

From More than One Right Zero then No Left Zero, it also follows that there is no left zero.

Also see

 * Element under Right Operation is Left Identity
 * All Elements of Left Operation are Left Zeroes