text: Semigroups and monoids If  is a semigroup or a monoid, one can still use the notion of a generating set  of .  is a semigroup/monoid generating set of  if  is the smallest semigroup/monoid containing .
  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  is said to be a semigroup generating set of  if each element of  is a finite sum of elements of . Similarly, a set  is said to be a monoid generating set of  if each non-zero element of  is a finite sum of elements of .
  For example, {1} is a monoid generator of the set of natural numbers . The set {1} is also a semigroup generator of the positive natural numbers . 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 integers , {1} is not a monoid generator of the set of integers. Indeed, the integer −1 cannot be expressed as a finite sum of 1s.