Left Inverse for All is Right Inverse

Theorem

Let $\struct {S, \circ}$ be a semigroup with a left identity $e_L$ such that:

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

That is, every element of $S$ has a left inverse with respect to the left identity.

Then $x \circ x_L = e_L$, that is, $x_L$ is also a right inverse with respect to the left identity.

Let $y = x \circ x_L$. Then:

$\ds e_L \circ y$

$=$

$\ds \paren {y_L \circ y} \circ y$

Definition of Left Inverse Element

$\ds $

$=$

$\ds y_L \circ \paren {y \circ y}$

Semigroup Axiom $\text S 1$: Associativity

$\ds $

$=$

$\ds y_L \circ y$

$\ds $

$=$

$\ds e_L$

Definition of Left Inverse Element

So $x \circ x_L = e_L$, and $x_L$ behaves as a right inverse as well as a left inverse with respect to the left identity.

$\blacksquare$

1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Semigroups: Exercise $5$