Monoid¶

Definition¶

A monoid is a set $A$ together with a binary operation $\star$ that is associative and has an identity element.

Interpretation¶

A monoid is a semigroup with a neutral element: a set with an associative operation and an identity element $e$. Inverses are not required, which is what separates a monoid from a group.

Associativity makes repeated products $a\star a\star\cdots\star a$ unambiguous and the identity gives the empty product a value, so powers $a^n$ are defined for every integer $n\ge0$$0$, with $a^0=e$$e$. Negative powers of $a$ are defined when $a$ has an inverse. A group guarantees this for every element; a monoid can have some invertible elements without being a group.

Typical monoids are $(\mathbb{N},+)$ with identity $0$ (so $0\in\mathbb{N}$$\mathbb{N}$, as in the Examples), $(\mathbb{Z},\cdot)$ with identity $1$, and the functions from a set $A$ to itself under composition, with the identity function $x\mapsto x$$x$. The first two are not groups: $1$ has no additive inverse in $\mathbb{N}$, and $2$ has no multiplicative inverse in $\mathbb{Z}$. The full self-map monoid is not a group when $A$ has at least two elements, because it then contains a non-bijective constant function. For an empty or one-element $A$, it is the trivial group. The invertible self-maps, the permutations of $A$, always form a group.

Key Results¶

• Every group is a monoid.
• Associativity, identity elements, and inverse elements determine which algebraic structure is present.

Examples¶

• $(\mathbb{N},+)$ is a monoid with identity $0$.

Prerequisites¶

• Semigroup
• Identity Element

Related Concepts¶

• Group
• Binary Operation
• Algebraic Structure
• Associativity

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: semigroup with identity Symbols: element of, integers, natural numbers

Last updated: 27 October 2026 11:52