Element under Right Operation is Left Identity

Theorem

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

Then all of the elements of $\struct {S, \to}$ left identities.

Proof

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

From the definition of right operation:

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

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

$\blacksquare$

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 $4$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.3 \ \text{(b)}$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Semigroups: Exercise $3 \ \text{(i)}$