|
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, ... (Sloane's A027851).
The number of semigroups of order  , 2, ... with
one idempotent are 1, 2, 5, 19,
132, 3107, 623615, ... (Sloane's A002786), and with two idempotents
are 2, 7, 37, 216, 1780, 32652, ... (Sloane's A002787). The number  of semigroups
having  , 3, ... idempotents
are 1, 2, 6, 26, 135, 875, ... (Sloane's A002788).
Birget, J.-C.; Margolis, S.; Meakin, J. and Sapir, M. (Eds.). Algorithmic Problems in Groups and Semigroups. Boston,
MA: Birkhäuser, 2000.
Clifford, A. H. and Preston, G. B. The Algebraic Theory of Semigroups. Providence, RI: Amer.
Math. Soc., 1961.
Howie, J. H. Fundamentals of Semigroup Theory. Oxford, England: Oxford
University Press, 1996.
Lallement, G. Semigroups and Combinatorial Applications. New York: Wiley,
1979.
Sloane, N. J. A. Sequences A001423/M3550, A002786/M1522, A002787/M1802, A002788/M1679, A027851, and A058131 in "The On-Line Encyclopedia of Integer Sequences."
|
Contact the MathWorld Team
