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Definition and examples

A basis is a subset B {\displaystyle B} of the elements of S {\displaystyle S} with the property that every element of S {\displaystyle S} may be formed in a unique way by choosing finitely many basis elements b i {\displaystyle b_{i}} of B {\displaystyle B}, choosing a nonzero integer k i {\displaystyle k_{i}} for each of the chosen basis elements, and adding together k i {\displaystyle k_{i}} copies of the basis elements b i {\displaystyle b_{i}} for which k i {\displaystyle k_{i}} is positive, and − k i {\displaystyle -k_{i}} copies of − b i {\displaystyle -b_{i}} for each basis element for which k i {\displaystyle k_{i}} is negative. As a special case, the identity element can always be formed in this way as the combination of zero basis elements, according to the usual convention for an empty sum, and it must not be possible to find any other combination that represents the identity.