All Elements of Left Operation are Left Zeroes

Theorem

Let $\struct {S, \leftarrow}$ be an algebraic structure in which the operation $\leftarrow$ is the left operation.

Then no matter what $S$ is, $\struct {S, \leftarrow}$ 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 Structure under Left Operation is Semigroup that $\struct {S, \leftarrow}$ 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.

$\blacksquare$

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

Also see

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

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $6$