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 can be empty. The total number of semigroups of order are 1, 4, 18, 126, 1160, 15973, 836021, ... (Sloane's A001423). The number of semigroups of order 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 Idempotents are 1, 2, 6, 26, 135, 875, ... (Sloane's A002788).

**References**

Clifford, A. H. and Preston, G. B. *The Algebraic Theory of Semigroups.* Providence, RI: Amer. Math. Soc., 1961.

Sloane, N. J. A. Sequences
A001423/M3550,
A002786/M1522,
A002787/M1802, and
A002787/M1679
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-26