Semigroup

A semigroup is an algebraic structure consisting of a set $S$ and a binary operation $*$ that satisfies the associative property. In more precise terms, a semigroup is a pair $(S, *)$ such that:

1. $S$ is a non-empty set.
2. $*$ is a binary operation on $S$, meaning that for any two elements $a, b \in S$, the result of $a * b$ is also an element in $S$.
3. The associative property holds for the binary operation, which means that for any elements $a, b, c \in S$, the equation $(a * b) * c = a * (b * c)$ is true.

It's important to note that a semigroup does not require the presence of an identity element or the existence of inverse elements, unlike other algebraic structures like groups or monoids. Semigroups appear in various areas of mathematics, such as algebra, automata theory, and the study of partial differential equations.