Semigroup

Definition

A semigroup is a pair $(S, \ast)$, where $S$ is a set and $\ast$ is a binary operation on $S$ that is associative. Thus $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in S$.

Remarks

The positive integers with addition form a semigroup that has no identity element. Each monoid and each group is a semigroup. Some authors allow the empty set as a semigroup, and others do not.

Explanations

L1 human

A semigroup is the simplest structure with an associative operation. You can combine elements and the grouping does not matter, but maybe nothing acts as "do nothing" and no step can be undone. An example is the set of nonempty words over an alphabet, with the operation that writes one word after the other.