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A numerical semigroup is a subset of the nonnegative integers that contains zero, is closed under addition, and has finite complement within the nonnegative integers. For example,Its missing values are $1,2,4,7$. To verify that no larger value is missing, observe that $8,9,10$ are present and then repeatedly add three, using consecutive-base induction for additive semigroups.

A finite family of positive integer generators produces a numerical semigroup exactly when their greatest common divisor is one. The condition is necessary because all generated integers are multiples of that greatest common divisor. Sufficiency follows by finding a generated representative of each residue class modulo one generator and then adding multiples of that generator.