Group

A set $S$ equipped with a binary operation $S\times S \rightarrow S$, which satisfies the following three axioms:

• Associativity : For all $a,b$ and $c$ in $S$, the equation $(a\cdot, b) \cdot c = a \cdot (b \cdot c)$ holds
• Identity Element : There exists an element $e$ in $S$ such that $e\cdot a = a \cdot e = a $ holds for all $a\in S$.
• Inverse Element : There exists an unique element $b$ in $S$ such that $a \cdot b = b \cdot a = e $ holds for all $a\in S$.

Semigroup

A set $S$ equipped with a binary operation $S\times S \rightarrow S$, which satisfies the following axiom:

• Associativity : For all $a,b$ and $c$ in $S$, the equation $(a\cdot, b) \cdot c = a \cdot (b \cdot c)$ holds

Monoid

Monoid is a semigroup with identity element.

A set $S$ equipped with a binary operation $S\times S \rightarrow S$, which satisfies the following two axioms:

• Associativity : For all $a,b$ and $c$ in $S$, the equation $(a\cdot, b) \cdot c = a \cdot (b \cdot c)$ holds
• Identity Element : There exists an element $e$ in $S$ such that $e\cdot a = a \cdot e = a $ holds for all $a\in S$.