Period of a subadditive function
Definition
Let \(\mathbb{Q}_0^+\) denote the set of non-negative rational numbers, let \(f : \mathbb{N} \to \mathbb{Q}_0^+\) be a subadditive function and \(m \in \mathbb{N} \setminus \{0\}\). It is said that \(f\) has period \(m\) if for all \(x \in \mathbb{N}\), it holds \(f(x + m) = f(x)\). It is denoted by \(\mathscr{JF}_m\) the set of \(m-\)periodic subadditive functions.
Given a subadditive function \(f\), the set
\[ M(f) = \{x \in \mathbb{N} ~ | ~ f(x) \le x\}, \]
is a submonoid of \(\mathbb{N}\). Moreover, if \(f\) has period \(m \in \mathbb{N} \setminus \{0\}\), it can be proven that \(M(f)\) is a numerical semigroup.
Examples
\(\circ\) Let \(f: \mathbb{N} \to \mathbb{Q}_0^+\) such that \(f(0) = 0, f(1) = 7, f(2) = 4\) and for all \(x \in \mathbb{N}\) such that \(x \ge 3\), \(f(x) = f(x ~ mod ~3 )\). Let us prove that \(f\) is a \(3-\)periodic subadditive function. Let \(x, y \in \mathbb{N}\), we divide in two cases.
\(x = y\): we have that \(f(1 + 1) = f(2) = 4 < 14 = f(1) + f(1)\) and \(f(2 + 2) = f(4) = f(1) = 7 < 8 = f(2) + f(2)\). If \(x \equiv 1 ~ (mod ~ 3)\), \(f(2x) = f(2) < f(1) + f(1) = f(x) + f(x)\) and if \(x \equiv 2 ~ (mod ~ 3)\), \(f(2x) = f(4) < f(2) + f(2) = f(x) + f(x)\).
\(x \equiv 1 ~ (mod ~ 3), y \equiv 2 ~ (mod ~ 3)\), then \(f(x + y) = f(0) = 0 < f(x) + f(y)\).
In conclusion. \(f\) is a \(3-\)periodic subadditive function.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(f\) be a periodic subadditive function with period \(m\). Given a list of images of the integers from \(1\) to \(m\), the function NumericalSemigroupBySubAdditiveFunction returns the numerical semigroup \(M(f)\) if it is possible (that means, if the list generates a subadditive function). The image of \(m\) has to be \(0\).
gap> S := NumericalSemigroupBySubAdditiveFunction([6, 5, 3, 0]);
<Numerical semigroup>\(S\) is the numerical semigroup of the subadditive function \(f\) such that \(f(0) = 0, f(1) = 6, f(2) = 5, f(3) = 3\) and period \(m = 4\). The function SmallElements returns a list with the left elements and the conductor of the numerical semigroup.
gap> SmallElements(S);
[ 0, 3, 4, 6 ]To sum up, the numerical semigroup generated by \(f\) is \(S = M(f) = \{0, 3, 4, 6, \rightarrow\}\).
References
https://gap-packages.github.io/ numericalsgps.