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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.3 \ \text{(b)}$