Element under Right Operation is Left Identity

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Theorem

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


Then $\struct {S, \rightarrow}$ is a semigroup all of whose elements are left identities.


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


Proof

From Right Operation is Associative, $\rightarrow$ is associative.

By the nature of the right operation, $\struct {S, \rightarrow}$ is closed:

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

whatever $S$ may be.

So $\struct {S, \rightarrow}$ is a semigroup.


From the definition of right operation:

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

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

$\blacksquare$


From More than one Left Identity then no Right Identity, it also follows that there is no right identity.


Also see


Sources