Semigroup¶

Definition¶

A semigroup is a set $A$ together with an associative binary operation $\star$.

Interpretation¶

A semigroup is the simplest algebraic structure with a rule for computing: a set with a binary operation that is associative, $(a\star b)\star c=a\star(b\star c)$$c)$. Associativity makes a repeated product such as $a\star b\star c\star d$ independent of where the parentheses are placed, so it can be written without them. It does not allow the factors to be reordered.

Nothing else is required: no identity element and no inverses. The positive integers $\{1,2,3,\ldots\}$ under addition form a semigroup without identity, since $0$ is missing. Adding an identity gives a monoid, and adding inverses as well gives a group. Subtraction on $\mathbb{Z}$ is a binary operation that does not form a semigroup: $(5-3)-1=1$$1$, but $5-(3-1)=3$$3$.

Key Results¶

• Every monoid is a semigroup.
• Every group is a semigroup.

Examples¶

• $(\mathbb{N},+)$ is a semigroup.

Prerequisites¶

• Associativity
• Binary Operation

Related Concepts¶

• Group
• Algebraic Structure

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: associative magma Symbols: integers, natural numbers

Last updated: 20 October 2026 03:02