Monoid

Definition

A monoid is an algebraic structure, $(A,\ast ,1)$ where,

• $A:\mathrm{Set}$ is the carrier.
• $\ast :A\to A\to A$ is a binary operator.
• $1:A$ is the unit element.
• $x\ast 1=1\ast x=x$ - $1$ is a unit over $\ast$.
• $x\ast (y\ast z)=(x\ast y)\ast z$ - $\ast$ is associative.