Monoids

Definition: Monoid

A monoid $(M,v)$ is a set $M$ equipped with an operation $v:M\times M\to M$ which satisfies the following:

• Associativity: $v(v(a,b),c)=v(a,v(b,c))$ for all $a,b,c\in M$.
• Existence of a neutral element: There is some $e\in M$ such that $v(e,a)=v(a,e)=a$ for all $a\in M$.

Theorem: Monoids are Semigroups

Every monoid is a semigroup.

PROOF

TODO