Inverse in Monoid is Unique

Theorem

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

Then an element $x \in S$ can have at most one inverse for $\circ$.

Proof

Let $e$ be the identity element of $\struct {S, \circ}$.

Suppose $x \in S$ has two inverses: $y$ and $z$.

Then:

$\ds y$

$=$

$\ds y \circ e$

Definition of Identity Element

$\ds $

$=$

$\ds y \circ \paren {x \circ z}$

Definition of Inverse Element

$\ds $

$=$

$\ds \paren {y \circ x} \circ z$

Monoid Axiom $\text S 1$: Associativity

$\ds $

$=$

$\ds e \circ z$

Definition of Inverse Element

$\ds $

$=$

$\ds z$

Definition of Identity Element

Similarly:

$\ds y$

$=$

$\ds e \circ y$

Definition of Identity Element

$\ds $

$=$

$\ds \paren {z \circ x} \circ y$

Definition of Inverse Element

$\ds $

$=$

$\ds z \circ \paren {x \circ y}$

Monoid Axiom $\text S 1$: Associativity

$\ds $

$=$

$\ds z \circ e$

Definition of Inverse Element

$\ds $

$=$

$\ds z$

Definition of Identity Element

So whichever way round you do it, $y = z$ and the inverse of $x$ is unique.

$\blacksquare$

Also see

Sources

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Theorem $4.2$
1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups: Theorem $1.2 \text{(iii)}$
1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $2$: Examples of Groups and Homomorphisms: $2.2$ Definitions $\text{(ii)}$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 31.2$ Identity element and inverses
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids
1999: J.C. Rosales and P.A. García-Sánchez: Finitely Generated Commutative Monoids ... (previous) ... (next): Chapter $1$: Basic Definitions and Results