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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 $n$ are 1, 4, 18, 126, 1160, 15973, 836021, ... (Sloane's A001423). The number of semigroups of order $n$ 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 $a(n)$ of semigroups having $n$ Idempotents are 1, 2, 6, 26, 135, 875, ... (Sloane's A002788).

See also Associative, Binary Operator, Free Semigroup, Groupoid, Inverse Semigroup, Monoid, Quasigroup


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.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

© 1996-9 Eric W. Weisstein