Element under Left Operation is Right Identity

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Theorem

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


Then all of the elements of $\struct {S, \gets}$ are right identities.


Proof

From Structure under Left Operation is Semigroup, $\struct {S, \gets}$ is a semigroup.


From the definition of left operation:

$\forall x, y \in S: x \gets y = x$

from which it is apparent that all elements of $S$ are right identities.

$\blacksquare$


Also see


Sources