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Semigroup

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).

See also

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

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References

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."

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Semigroup

Cite this as:

Weisstein, Eric W. "Semigroup." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Semigroup.html

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