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). The number of semigroups of order n = 1, 2, ... with one idempotent are 1, 2, 5, 19, 132, 3107, 623615, ... (OEIS A002786), and with two idempotents are 2, 7, 37, 216, 1780, 32652, ... (OEIS A002787). The number a(n) of semigroups having n = 2, 3, ... idempotents are 1, 2, 6, 26, 135, 875, ... (OEIS A002788).

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