Monoid

Definition

A monoid is a triple $(M, \ast, e)$, where $(M, \ast)$ is a semigroup and $e \in M$ is an identity element for $\ast$. Thus $(a \ast b) \ast c = a \ast (b \ast c)$ and $e \ast a = a \ast e = a$ for all $a, b, c \in M$.

Remarks

The natural numbers $\mathbb{N} = \{0, 1, 2, \ldots\}$ with addition form a monoid with identity $0$. The integers with multiplication form a monoid with identity $1$. The identity element of a monoid is unique, thus some authors write a monoid as the pair $(M, \ast)$. Each group is a monoid in which each element has an inverse.

Explanations

L1 human

A monoid is a set with an associative operation and an element that does nothing. The strings of letters with concatenation form a monoid, and the empty string is the identity. In a monoid, an element need not have an inverse: you cannot undo the addition of a letter.