Archimedean semigroup
Let be a commutative semigroup. We say an element divides an element , written , if there is an element such that .
An Archimedean semigroup is a commutative semigroup with the property that for all there is a natural number such that .
This is related to the Archimedean property of positive real numbers : if then there is a natural number such that . Except that the notation is additive rather than multiplicative, this is the same as saying that is an Archimedean semigroup.