Definition:Identity (Abstract Algebra)/Left Identity

Definition

Let $\struct {S, \circ}$ be an algebraic structure.

An element $e_L \in S$ is called a left identity if and only if:

$\forall x \in S: e_L \circ x = x$

Also known as

• Left neutral element
• Left unit

Also see

• Definition:Right Identity
• Definition:Identity Element

• Results about identity elements can be found here.

Sources

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.4: \ 9$
• 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 $4.3$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Operations
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Semigroups: Exercise $3$