The author considers sets of integer vectors containing the zero vector and closed under addition, the integral monoids, and provides conditions under which they contain a finite subset of integer vectors which generate the entire monoid as non-negative integer combinations. It is also shown that integral monoids, which have a finite subset of the type just described, can be represented as the sum of a group and a monoid whose convex span is a pointed cone. The paper concludes with some applications to the theory of integer programming.
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