Internal numerical semigroups

Abstract: In this paper the tree structure of numerical semigroups is studied. An internal numerical semigroup is a semigroup located in an internal node of the tree. Analogous a leaf numerical semigroup is placed in a leaf node. Internal semigroups with fixed multiplicity, Frobenius number or genus are studied by providing algorithms to construct all of them. Several conjectures are established, for example, in each case (fixed multiplicity, fixed Frobenius number and fixed genus respectively), the results suggest that there are always more internal than leaf numerical semigroups. Finally, numerical semigroups with fixed multiplicity and Frobenius number simultaneously are investigated. In this case with two invariants fixed, moreover closed formulas to count the number of internal and leaf semigroups are provided for some values of multiplicity and Frobenius number.

• The paper introduces the internal/leaf decomposition of the numerical-semigroup tree and proves that counting genus-4 semigroups reduces to internal semigroups of genus4.
• It establishes Frobenius variety, covariety, pseudo-variety, and ratio-covariety structures that support tree-based enumeration under fixed genus, Frobenius number, multiplicity, or paired invariants.
• For F/2 < m < F, the paper proves that exactly 2^{F-m-1} internal semigroups have multiplicity m and Frobenius number F, while leaves do not occur, whereas broader monotonicity and golden-ratio growth claims remain conjectural.

The internal/leaf dichotomy in the tree of numerical semigroups

A numerical semigroup $S$ is a submonoid of $(\mathbb{N},+)$ with finite complement, and its fundamental invariants are the multiplicity $m(S)$, Frobenius number $F(S)$, and genus $g(S)$. The paper of Casas, Madrid, and Rosales (2608.19984) introduces a classification of numerical semigroups based on their position in the well-known tree $G(\mathscr{S})$, whose root is $\mathbb{N}$ and whose edges connect a semigroup $S$ to its children $S\setminus\{x\}$ for $x \in \mathrm{msg}(S)$ with $(\mathbb{N},+)$0. A semigroup is internal if $(\mathbb{N},+)$1—equivalently, it has children in the tree—and leaf otherwise. Writing $(\mathbb{N},+)$2 and $(\mathbb{N},+)$3 for these two classes, the authors establish the key identity

$(\mathbb{N},+)$4

so that counting semigroups of genus $(\mathbb{N},+)$5 reduces entirely to enumerating internal semigroups of genus $(\mathbb{N},+)$6. This reframes the classical problem of computing $(\mathbb{N},+)$7 (the subject of extensive work on the Bras-Amorós monotonicity conjecture $(\mathbb{N},+)$8) in terms of the internal class.

Fixed genus: a Frobenius variety and Fibonacci-like growth

The central structural observation is that $(\mathbb{N},+)$9 is a Frobenius variety: it is closed under finite intersection and under the operation $m(S)$0. This permits application of the general theory of Frobenius varieties to conclude that $m(S)$1 is itself a tree rooted at $m(S)$2, whose vertices at depth $m(S)$3 are exactly $m(S)$4. Since every genus is realized by an internal semigroup (e.g., $m(S)$5), an explicit breadth-first algorithm computes all of $m(S)$6.

The computational output up to genus 26 supports three conjectures: that both $m(S)$7 and $m(S)$8 are nondecreasing in $m(S)$9, and that $F(S)$0 for all $F(S)$1. At genus 26 the counts are $F(S)$2 versus $F(S)$3. More strikingly, the data exhibit Fibonacci-like behavior analogous to that observed by Bras-Amorós for $F(S)$4: the ratios $F(S)$5 and $F(S)$6 approach 1, while $F(S)$7 and $F(S)$8 appear to converge to the golden ratio $F(S)$9. The resulting conjecture ($g(S)$0 for $g(S)$1, and the leaf analogue for $g(S)$2) strictly implies the two monotonicity conjectures. These claims rest entirely on empirical evidence; no proof is given.

Fixed Frobenius number: a covariety

For fixed positive integer $g(S)$3, the family $g(S)$4 is shown to be a covariety with minimum $g(S)$5, closed under intersection and under removal of the multiplicity when the result differs from the minimum. Consequently $g(S)$6 is a tree rooted at $g(S)$7, and children of a vertex are obtained by adjoining special gaps smaller than the multiplicity. An algorithm exploiting Apéry sets with respect to $g(S)$8 enumerates the entire class; for example, $g(S)$9 consists of four semigroups.

Computation up to $G(\mathscr{S})$0 yields counts growing roughly geometrically (e.g., $G(\mathscr{S})$1 against $G(\mathscr{S})$2) and motivates two further conjectures: $G(\mathscr{S})$3 for all $G(\mathscr{S})$4, and the oscillatory pattern $G(\mathscr{S})$5 for odd $G(\mathscr{S})$6. Both are unproven.

Fixed multiplicity: a Frobenius pseudo-variety

For $G(\mathscr{S})$7, the family $G(\mathscr{S})$8 is a Frobenius pseudo-variety with maximum $G(\mathscr{S})$9, so its associated graph is a tree rooted at $\mathbb{N}$0 with children obtained by deleting minimal generators other than $\mathbb{N}$1 that exceed the Frobenius number. The attainable genera are exactly $\mathbb{N}$2, witnessed by an explicit construction. The enumeration algorithm produces tables up to genus 26 for each multiplicity, from which several refined conjectures emerge:

• $\mathbb{N}$3 for $\mathbb{N}$4;
• monotonicity of internal counts holds for $\mathbb{N}$5 and of leaf counts for $\mathbb{N}$6 — notably, the analogues fail for small $\mathbb{N}$7: $\mathbb{N}$8 and $\mathbb{N}$9;
• $S$0 whenever $S$1.

These threshold phenomena indicate that the global monotonicity behavior is genuinely multiplicity-dependent, and the small-$S$2 counterexamples delimit the scope of any future proof.

Fixed multiplicity and Frobenius number: closed formulas

Fixing both invariants, $S$3 (with $S$4 and $S$5) is proved to be a ratio-covariety with minimum $S$6, again yielding a tree and an enumeration algorithm via special gaps strictly between $S$7 and the ratio $S$8.

The strongest result of the paper is a closed formula: for $S$9 and $S\setminus\{x\}$0,

$S\setminus\{x\}$1

The proof is elementary: since $S\setminus\{x\}$2, every such semigroup has the form $S\setminus\{x\}$3 plus an arbitrary subset of $S\setminus\{x\}$4, and $S\setminus\{x\}$5 always remains a minimal generator exceeding $S\setminus\{x\}$6, forcing internality. Hence, in this regime, all numerical semigroups with the given invariants are internal nodes, and the total count equals $S\setminus\{x\}$7. This is consistent with the general conjecture $S\setminus\{x\}$8 supported by the tabulated data (e.g., for $S\setminus\{x\}$9, where multiplicities $x \in \mathrm{msg}(S)$0 contribute only internal semigroups).

One subtlety deserves mention: when the tree $x \in \mathrm{msg}(S)$1 consists of a single node (as for $x \in \mathrm{msg}(S)$2, $x \in \mathrm{msg}(S)$3), the internal/leaf terminology refers exclusively to the position in the full tree $x \in \mathrm{msg}(S)$4, not to the sub-tree—a distinction the authors make explicit.

Limitations and open questions

All comparative statements beyond the closed formula of the previous section are conjectural, supported only by computation up to genus 26 and Frobenius number 56. In particular, the following remain open:

• the monotonicity conjectures for $x \in \mathrm{msg}(S)$5, $x \in \mathrm{msg}(S)$6, and their fixed-multiplicity and fixed-Frobenius analogues;
• the Fibonacci-type asymptotics, including convergence of the growth ratios to $x \in \mathrm{msg}(S)$7;
• the dominance conjectures $x \in \mathrm{msg}(S)$8 in each fixed-invariant setting;
• whether the Fibonacci-like inequalities extend to the fixed-Frobenius case, for which no analogue is proposed.

The algorithms are exponential-time tree traversals, so extending the computational range substantially may require structural insights rather than raw enumeration.

Conclusion

This paper introduces the internal/leaf decomposition of the tree of numerical semigroups, proves that each of the families $x \in \mathrm{msg}(S)$9, $(\mathbb{N},+)$00, $(\mathbb{N},+)$01, and $(\mathbb{N},+)$02 carries appropriate tree structure (via the Frobenius variety, covariety, pseudo-variety, and ratio-covariety frameworks respectively), and supplies algorithms enumerating each class. The reduction $(\mathbb{N},+)$03 ties the classification directly to the classical genus-counting problem, and the closed formula $(\mathbb{N},+)$04 in the large-multiplicity regime provides the first exact counts within this framework. The extensive conjectural layer—monotonicity, golden-ratio asymptotics, and internal-over-leaf dominance—constitutes a concrete agenda for subsequent work.