Monoid

DEF-MO. Monoid.

A monoid is a unital semigroup.

PROP-MO-U.

A monoid is uniquely unital.

Proof.Let $M$ be a monoid. If both $e$ and $e’$ are neutral elements of $M$,

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PROP-MO-LR.

Let $M$ be a monoid and $a\in M$. $a$ is left-invertible if and only if $a$ is right-invertible.

Proof.Let $e$ be the neutral element.

• If $a$ is left-invertible, then there is $x\in M$ such that $xa=e$. We have

  \[x(ax) = (xa)x = ex = x,\]

  which follows that $ax=e$.

• If $a$ is right-invertible, then there is $x\in M$ such that $ax=e$. We have

  \[(xa)x = x(ax) = xe = x,\]

  which follows that $xa=e$.

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