that, for instance, There is no other way to write in the same basis. However, with a different basis such as it can be written as Generalizing this example, every lattice forms a finitely-generated free abelian group. The integer lattice has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if is a integer matrix with then the rows of form a basis, and conversely every basis of the integer lattice has this form. For more on the two-dimensional case, see fundamental pair of periods. Constructions Every set can be the basis of a free abelian group, which is unique up to group isomorphisms. The free abelian group for a given basis set can be constructed in several different but equivalent ways: as a direct sum of copies of the integers, as a family of integer-valued functions, as a signed multiset, or by a presentation of a group. Products and sums The direct product of groups consists of tuples of an element from each group in the product, with componentwise addition. The direct product of two free abelian groups is itself free abelian, with basis the disjoint union of the bases of the two groups. More generally the direct product of any finite number of free abelian groups is free abelian. The integer lattice, for instance, is isomorphic to the direct product of copies of the integer The trivial group is also considered to be free abelian, with basis the empty set. It may be interpreted as an empty product, the direct product of zero copies For infinite families of free abelian groups, the direct product is not necessarily free abelian. For instance the Baer–Specker group an uncountable group formed as the direct product of countably many copies was shown in 1937 by Reinhold Baer to not be free abelian, although Ernst Specker proved in 1950 that all of its countable subgroups are free abelian. Instead, to obtain a free abelian group from an infinite family of groups, the direct sum rather than the direct product should be used. The direct sum and direct product are the same when they are applied to finitely many groups, but differ on infinite families of groups. In the direct sum, the elements are again tuples of elements from each group, but with the restriction that all but finitely many of these elements are the