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 need inverses and exist only in 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$. None of these is a group: in $(\mathbb{Z},\cdot)$ the element $2$ has no inverse, and a function that is not bijective cannot be undone. The invertible functions, the permutations of $A$, do 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: 20 October 2026 03:02