# Numerical semigroup

## Definition

A **numerical semigroup** is defined as a submonoid of the monoid of nonnegative integers under addition, whose set-theoretic complement is finite. In other words, it is a subset of the nonnegative integers containing zero, closed under addition, and containing all but finitely many positive integers.

A **minimal system of generators** is a minimal subset of the numerical semigroup such that the given numerical semigroup is precisely the set of nonnegative integer combinations of elements from the subset. Every numerical semigroup has a unique minimal system of generators.

## Facts

If the gcd of all elements from a submonoid of the natural numbers is 1, then it must be a numerical semigroup.

The key idea behind the proof is for the case of a numerical semigroup with two generators: the so-called postage stamp problem.