identity for their group. The direct sum of infinitely many free abelian groups remains free abelian. It has a basis consisting of tuples in which all but one element is the identity, with the remaining element part of a basis for its group. Every free abelian group may be described as a direct sum of copies with one copy for each member of its basis. This construction allows any set to become the basis of a free abelian group. Integer functions and formal sums Given a one can define a group whose elements are functions from to the integers, where the parenthesis in the superscript indicates that only the functions with finitely many nonzero values are included. If and are two such functions, then is the function whose values are sums of the values in that is, This pointwise addition operation gives the structure of an abelian group. Each element from the given set corresponds to a member the function for which and for which for Every function in is uniquely a linear combination of a finite number of basis elements: Thus, these elements form a basis and is a free abelian group. In this way, every set can be made into the basis of a free abelian group. The elements of may also be written as formal sums, expressions in the form of a sum of finitely many terms, where each term is written as the product of a nonzero integer with a distinct member These expressions are considered equivalent when they have the same terms, regardless of the ordering of terms, and they may be added by forming the union of the terms, adding the integer coefficients to combine terms with the same basis element, and removing terms for which this combination produces a zero coefficient. They may also be interpreted as the signed multisets of finitely many elements Presentation A presentation of a group is a set of elements that generate the group (meaning that all group elements can be expressed as products of finitely many generators), together with "relators", products of generators that give the identity element. The elements of a group defined in this way are equivalence classes of sequences of generators and their inverses, under an equivalence relation that allows inserting or removing any relator or generator-inverse pair as a contiguous subsequence. The free abelian group with basis has a presentation in which the