Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. Mathematics A set for which there is a binary operation that is closed and associative.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. Any set for which there is a binary operation that is both closed and associative.
Etymologies
Examples

A semigroup is a set together with an associative operation, called the multiplication of the semigroup and notated xy rather than f (x, y).

The final equation for the evolution at the ensemble level is of the quantum dynamical semigroup type and has a structure extremely similar to the final one of the GRW theory.

X, whence the semigroup of all functions on a set X forms a monoid.

The semigroup of all nonempty words over a given alphabet has as a subsemigroup the words of even length; however the words of odd length do not form a subsemigroup because the concatenation of two oddlength words is not of odd length.

A commutative semigroup is a semigroup whose operation is commutative.

"Nonbreaking error handling is just an applicative functor on a partially applied disjoint union type constructor with semigroup error elements so what's the big deal?"

These considerations illustrate how the commonly used semigroup ansatz for time evolution of the unstable systems arises.

Th ... abc on adelic BostConnes unknown: But then, we can form the crystalline semigroup graded skewgroup algebra.
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