Editing Generating set of a group (section)

==Semigroups and monoids==
If <math>G</math> is a [[semigroup]] or a [[monoid]], one can still use the notion of a generating set <math>S</math> of <math>G</math>. <math>S</math> is a semigroup/monoid generating set of <math>G</math> if <math>G</math> is the smallest semigroup/monoid containing <math>S</math>. 

The definitions of generating set of a group using finite sums, given above, must be slightly modified when one deals with semigroups or monoids. Indeed, this definition should not use the notion of inverse operation anymore. The set <math>S</math> is said to be a semigroup generating set of <math>G</math> if each element of <math>G</math> is a finite sum of elements of <math>S</math>. Similarly, a set <math>S</math> is said to be a monoid generating set of <math>G</math> if each non-zero element of <math>G</math> is a finite sum of elements of <math>S</math>.

For example, {1} is a monoid generator of the set of [[natural number]]s <math>\N</math>. The set {1} is also a semigroup generator of the positive natural numbers <math>\N_{>0}</math>. However, the integer 0 can not be expressed as a (non-empty) sum of 1s, thus {1} is not a semigroup generator of the natural numbers. 

Similarly, while {1} is a group generator of the set of [[integer]]s <math>\mathbb Z</math>, {1} is not a monoid generator of the set of integers. Indeed, the integer −1 cannot be expressed as a finite sum of 1s.