# All Elements of Right Operation are Right Zeroes

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## 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 Structure under Right Operation is Semigroup 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.

$\blacksquare$

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

## Also see

## Sources

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