Monoid

A monoid is a non-empty set $S$ together with a binary operation on $S$, denoted as "$\cdot$", such that the following two requirements are satisfied

Associativity For all $a,b,c\in S$, one has $(a\cdot b)\cdot c=a\cdot (b\cdot c)$

Identity element There exists an identity element $e$ in $S$ such that, for every $a$ in $S$, one has $e\cdot a=a,a\cdot e=a$