Monoid

Definition of a Monoid

Given a [Binary Operation] $\ast$ : $S\times S\to S$ on a set $S$, the pair $(S,\ast )$ is called a monoid if it satisfies the following conditions:

1. $\ast$ is associative.
2. There exists an identity element $e\in S$.

Examples

1. $(\mathbb{Z},\times )$ is a Monoid. However, it is not a Group.
2. $(Ma{t}_{2\times 2}(\mathbb{R}),\times )$ is a Monoid. However, it is not a Group.