Identity of Monoid is Unique

Theorem

Let $\struct {S, \circ}$ be a monoid.

Then $S$ has a unique identity.

Proof

As $\struct {S, \circ}$ is an algebraic structure, the result Identity is Unique can be applied directly.

$\blacksquare$

Also see

• Inverse in Monoid is Unique
• Identity of Group is Unique

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \beta$
• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $2$: Examples of Groups and Homomorphisms: $2.2$ Definitions $\text{(i)}$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 31.1$ Identity element and inverses
• 1999: J.C. Rosales and P.A. García-Sánchez: Finitely Generated Commutative Monoids ... (previous) ... (next): Chapter $1$: Basic Definitions and Results