A semigroup is 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 element 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).
For the following counts, semigroups are identified when they are isomorphic or one is isomorphic to the other with its multiplication
reversed (Jürgensen and Wick 1977, pp. 69-71). The numbers of semigroups
of orders 
,
2, ... with exactly one idempotent are 1, 2, 5, 19,
132, 3107, 623615, ... (OEIS A002786). For
orders 
, 3, ..., the corresponding numbers
with exactly two idempotents are 2, 7, 37, 216, 1780,
32652, ... (OEIS A002787). The numbers of semigroups
of orders 
,
2, ... whose elements are all idempotents are 1, 2,
6, 26, 135, 875, ... (OEIS A002788).
." Semigroup Forum 14, 69-79, 1977.Lallement,
G.