A **numerical semigroup** is a special type of subset of the non-negative integers. Specifically, it is a subgroup of the non-negative integers under addition that is closed under addition and contains the identity element 0. More formally, a numerical semigroup is defined as follows: 1. It is a subset \( S \) of the non-negative integers \( \mathbb{N}_0 = \{0, 1, 2, \ldots\} \).
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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 . To verify that no larger value is missing, observe that 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.