Semigroup

Idea

A semigroup is an algebraic structure consisting of a set equipped with an associative binary operation. It generalizes the concept of a monoid by removing the requirement for an identity element.

Definition

A semigroup is a tuple $(S,\cdot )$ where $S$ is a set and $\cdot :S\times S\to S$ is a binary operation satisfying: \forall x, y, z \in S, (x \cdot y) \cdot z = x \cdot (y \cdot z) \tag{assoc}

Properties

• Every monoid is a semigroup.
• Every semigroup is a group.
• A semigroup $S$ is a monoid if and only if there exists an element $e:S$ such that $e\cdot x=x$ and $x\cdot e=x$ for all $x:S$.
• For any semigroup $S$, the unitization $S^1$ is formed by adjoining a new element $1$ and defining $1\cdot s=s=s\cdot 1$ for all $s\in S\cup \{1\}$, which forms a monoid.
• If $S$ is a semigroup, the set of functions $S\to S$ under composition also forms a semigroup.

References

• pinter2010-abstract-algebra