A monoid is a mathematical structure that consists of a set equipped with a binary operation. This operation must satisfy two key properties: it is associative, meaning the order in which operations are performed does not change the result, and it has an identity element, which is a special element in the set that, when combined with any other element, leaves it unchanged. For example, the set of natural numbers with the operation of addition forms a monoid. Here, the identity element is 0, since adding 0 to any number does not change its value. Other examples include strings with concatenation and matrices with multiplication.