# Semigroup

From Groupprops

This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.

View other such properties

## Definition

QUICK PHRASES: set with associative binary operation, group without identity element and inverses

### Symbol-free definition

A **semigroup** is a set equipped with an associative binary operation.

A semigroup need not have an identity element.

The definition of semigroup does not require it to be nonempty, so the empty set is a semigroup. However, some variants of the definition require a semigroup to be a *nonempty* set with an associative binary operation.