Semigroup

A semigroup is a set $S$ together with an associative binary operation $\cdot : S \times S \to S$, meaning that for all $a,b,c \in S$, $(a\cdot b)\cdot c \;=\; a\cdot(b\cdot c).$

Semigroups are the weakest common algebraic setting in which repeated products are unambiguous (associativity lets you omit parentheses). A monoid is a semigroup with an identity element, and a group is a monoid in which every element has an inverse.

Examples:

• $(\mathbb{N},+)$ is a semigroup (addition is associative).
• $(\mathbb{N}_{\ge 1},\times)$ is a semigroup (multiplication is associative).
• The set of all $n\times n$ real matrices under matrix multiplication is a semigroup.
• (Edge case) If one allows empty algebraic structures, the empty set has a unique binary operation $\,\emptyset\times\emptyset\to\emptyset\,$ and is a semigroup, but it cannot be a monoid (no identity element exists).