A mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. A semigroup is an associative groupoid. A semigroup with an identity is called a monoid. A semigroup can be empty. The numbers of nonisomorphic semigroups of orders 1, 2, ... are 1, 5, 24, 188, 1915, ... (OEIS A027851).

associative | binary operator | free semigroup | groupoid | inverse semigroup | monoid | quasigroup