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every abelian group a commutative monoid.
* Any semigroup may be turned into a monoid simply by adjoining an element not in and defining for all . This conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids.
** Thus, an idempotent monoid (sometimes known as find-first) may be formed by adjoining an identity element to the left zero semigroup

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