All Elements of Left Operation are Left Zeroes

Theorem
Let $\left({S, \leftarrow}\right)$ be an algebraic structure in which the operation $\leftarrow$ is the left operation.

Then no matter what $S$ is, $\left({S, \leftarrow}\right)$ is a semigroup all of whose elements are left zeroes.

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

Proof
It is established in Left Operation All Elements Right Identities that $\left({S, \leftarrow}\right)$ is a semigroup.

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

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

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

Also see

 * Left Operation All Elements Right Identities
 * Right Operation All Elements Right Zeroes