Semigroup and monoid

A semigroup is a pair $(S,\ast )$ consisting of a set $S$ and an associative binary operation $\ast$ on $S$ (Binary operation on a set; associativity, commutativity, and a subset closed under the operation).

A monoid is a triple $(M,\ast ,e)$ in which $(M,\ast )$ is a semigroup and $e\in M$ is a two-sided identity for $\ast$ (Left identity, right identity, and two-sided identity for a binary operation), that is,

$e\ast x=x=x\ast e\text{for every }x\in M\mathrm{.}$

By A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity a binary operation has at most one two-sided identity, so $e$ is determined by $(M,\ast )$ and may be called the identity of $M$; it is written $e$, or $e_M$ when several monoids are in play, and $1$ or $0$ in multiplicative or additive notation. For that reason a monoid is often written simply as $(M,\ast )$, or as $M$.

A semigroup or monoid is commutative (for monoids also called abelian) when its operation is commutative.

A subset $N\subseteq M$ is a submonoid when $e\in N$ and $N$ is closed under $\ast$; the restricted operation then makes $(N,\ast ,e)$ a monoid, associativity being inherited (Binary operation on a set; associativity, commutativity, and a subset closed under the operation).

• The definite article is earned, not assumed. Writing "the identity" presupposes uniqueness, and that is exactly what A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity supplies, from the two defining equations alone. This is the first of the two uniqueness obligations on this page; the second is uniqueness of inverses (In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided), which is what will license writing "the inverse".

• A monoid is data, not a property. $(M,\ast ,e)$ carries the identity as part of the structure. The uniqueness result says nothing is lost by suppressing it from the notation, and nothing is gained by keeping it.

• Every group is a monoid (Group and abelian group), and the invertible elements of a monoid form a group (The invertible elements of a monoid form a group under the restricted operation), so the two notions are tied together in both directions.