For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including zero) under addition form a monoid, as do the nonzero integers under multiplication ⁠ ( Z ∖ ∖ { 0 } , ⋅ ⋅ ) ⁠. Adjoining inverses of all elements of the monoid ( Z ∖ ∖ { 0 } , ⋅ ⋅ ) produces a group ⁠ ( Q ∖ ∖ { 0 } , ⋅ ⋅ ) ⁠, and likewise adjoining inverses to any (abelian) monoid ⁠ M ⁠ produces a group known as the Grothendieck group of ⁠ M ⁠.