Numerical Semigroup Tree

• Numerical Semigroup Tree is a hierarchical structure that organizes numerical semigroups as nodes graded by genus, with parent–child relations defined by operations on gaps and generators.
• It leverages key invariants such as the Frobenius number, type, and gap vectors to refine semigroup classification and analyze asymptotic behaviors in enumerative studies.
• The structure underpins computational methods and algorithm designs, including breadth-first approaches, RGD, and seeds-based techniques, to efficiently traverse and analyze semigroup families.

Searching arXiv for the primary paper and closely related work on numerical semigroup trees and ordinarization/infinite-chain variants. A numerical semigroup tree is a rooted combinatorial structure that organizes numerical semigroups through elementary operations on gaps and generators. In the standard construction, a numerical semigroup is a cofinite submonoid of the nonnegative integers, the root is $\mathbb N_0$, and the level of a node is its genus. The tree has become a central device for structural, enumerative, and computational work on numerical semigroups: it supports breadth-first and depth-first generation by genus, fixed-parameter refinements by Frobenius number or type, asymptotic analysis of degrees and infinite chains, and several alternative rooted structures such as ordinarization trees, quasi-ordinarization forests, fixed-Frobenius irreducible trees, and arithmetic-variety trees (Chappelon et al., 20 Jul 2025).

1. Classical genus-graded tree

A numerical semigroup $S$ is a subset of $\mathbb N_0$ such that $0\in S$, $S$ is closed under addition, and $\mathbb N_0\setminus S$ is finite. Its basic invariants include the genus $g(S)=|\mathbb N_0\setminus S|$, the Frobenius number $F(S)=\max(\mathbb N_0\setminus S)$, the multiplicity $m(S)=\min(S\setminus\{0\})$, the embedding dimension $e(S)$, and the Apéry set $S$0 (Chappelon et al., 20 Jul 2025).

The standard numerical semigroup tree has all numerical semigroups as nodes and $S$1 as root. If $S$2 has positive genus, its parent is $S$3, so adjoining the Frobenius number decreases the genus by $S$4. Conversely, if $S$5 is a semigroup and $S$6 is a minimal generator of $S$7 with $S$8, then $S$9 is a child of $\mathbb N_0$0; the removed generator becomes the Frobenius number of the child. Thus levels are indexed by genus, and every edge increases genus by $\mathbb N_0$1 (Chappelon et al., 20 Jul 2025).

In this setting the children are obtained by removing minimal generators to the right of the Frobenius number. Several papers call these effective generators or right generators. The degree, or efficacy, of a node is the number of such generators, hence the number of children. This local branching is finite, and the degree distribution itself has asymptotic content: if $\mathbb N_0$2 denotes the number of genus-$\mathbb N_0$3 semigroups with degree $\mathbb N_0$4, then for each fixed $\mathbb N_0$5 one has

$\mathbb N_0$6

where $\mathbb N_0$7 is the total number of genus-$\mathbb N_0$8 semigroups and $\mathbb N_0$9 is the golden ratio (O'Dorney, 2024).

The classical tree is also the ambient structure for several refined notions. Right generators may be strong or weak depending on whether removing one creates a new right generator $0\in S$0, and pseudo-ordinary nodes have a particularly rigid right-generator structure. These distinctions underlie fast traversal algorithms such as RGD and seeds-based methods (Bras-Amorós et al., 2019).

2. Type, pseudo-Frobenius numbers, and the type-representation

A major refinement of the tree replaces pure genus stratification by a genus-and-type stratification. For a numerical semigroup $0\in S$1, the pseudo-Frobenius numbers are

$0\in S$2

equivalently the maximal non-elements for the order $0\in S$3. The type is $0\in S$4, and $0\in S$5 (Chappelon et al., 20 Jul 2025).

The basic inequalities governing this representation are $0\in S$6 and

$0\in S$7

Special families sit at the extremal low-type end: symmetric semigroups have $0\in S$8, and pseudosymmetric semigroups have $0\in S$9. At the opposite extreme, the condition $S$0 is equivalent to each of the following: $S$1, $S$2, and

$S$3

for some integer $S$4; in that case $S$5 (Chappelon et al., 20 Jul 2025).

The type-representation of the numerical semigroup tree arranges each genus level into contiguous blocks of increasing type $S$6. This makes the effect of parent–child operations on type visible. If $S$7 has positive genus and $S$8 is its parent, then

$S$9

with equality if and only if $\mathbb N_0\setminus S$0. Consequently $\mathbb N_0\setminus S$1, with equality exactly in the same extremal case, and if $\mathbb N_0\setminus S$2 then $\mathbb N_0\setminus S$3 (Chappelon et al., 20 Jul 2025).

This perspective emphasizes that the classical tree is not merely genus-graded. It is also stratified by pseudo-Frobenius structure, and near the top-type region that structure becomes rigid enough to admit explicit binary encodings and stabilization theorems.

3. Gap vectors, diagonal stabilization, and conjectural unimodality

Let $\mathbb N_0\setminus S$4 be the number of numerical semigroups of genus $\mathbb N_0\setminus S$5 and type $\mathbb N_0\setminus S$6, and write $\mathbb N_0\setminus S$7. In the high-type regime, namely when $\mathbb N_0\setminus S$8 or equivalently $\mathbb N_0\setminus S$9, the relevant gaps are confined to the interval $g(S)=|\mathbb N_0\setminus S|$0. One then encodes a semigroup by a binary gap vector

$g(S)=|\mathbb N_0\setminus S|$1

where $g(S)=|\mathbb N_0\setminus S|$2 iff $g(S)=|\mathbb N_0\setminus S|$3, and defines the cotype

$g(S)=|\mathbb N_0\setminus S|$4

For $g(S)=|\mathbb N_0\setminus S|$5, the associated set $g(S)=|\mathbb N_0\setminus S|$6 is a numerical semigroup of genus $g(S)=|\mathbb N_0\setminus S|$7, and

$g(S)=|\mathbb N_0\setminus S|$8

In particular, on the diagonal $g(S)=|\mathbb N_0\setminus S|$9, stable gap vectors are precisely those with $F(S)=\max(\mathbb N_0\setminus S)$0 (Chappelon et al., 20 Jul 2025).

This yields a stabilization theorem. If $F(S)=\max(\mathbb N_0\setminus S)$1 denotes the set of stable gap vectors of length $F(S)=\max(\mathbb N_0\setminus S)$2 with $F(S)=\max(\mathbb N_0\setminus S)$3 zeros and $F(S)=\max(\mathbb N_0\setminus S)$4 ones, then for every $F(S)=\max(\mathbb N_0\setminus S)$5,

$F(S)=\max(\mathbb N_0\setminus S)$6

Hence the diagonal counts become constant in $F(S)=\max(\mathbb N_0\setminus S)$7 once $F(S)=\max(\mathbb N_0\setminus S)$8 is large enough relative to the offset $F(S)=\max(\mathbb N_0\setminus S)$9. The first stabilized values reported are as follows (Chappelon et al., 20 Jul 2025).

Diagonal	Stabilized count	Valid when
$m(S)=\min(S\setminus\{0\})$0	1	$m(S)=\min(S\setminus\{0\})$1
$m(S)=\min(S\setminus\{0\})$2	3	$m(S)=\min(S\setminus\{0\})$3
$m(S)=\min(S\setminus\{0\})$4	7	$m(S)=\min(S\setminus\{0\})$5
$m(S)=\min(S\setminus\{0\})$6	15	$m(S)=\min(S\setminus\{0\})$7
$m(S)=\min(S\setminus\{0\})$8	35	$m(S)=\min(S\setminus\{0\})$9
$e(S)$0	78	$e(S)$1
$e(S)$2	161	$e(S)$3
$e(S)$4	367	$e(S)$5
$e(S)$6	757	$e(S)$7
$e(S)$8	1632	$e(S)$9
$S$00	3436	$S$01

The stabilization has an explicit shift mechanism. If $S$02, then

$S$03

is a bijection from genus $S$04, type $S$05 semigroups to genus $S$06, type $S$07 semigroups, and in gap-vector language the vector $S$08 is unchanged. This is the structural source of the constant diagonal counts (Chappelon et al., 20 Jul 2025).

The same computations support several global conjectures. For every $S$09, the sequence

$S$10

is unimodal, with the peak typically near $S$11, and the conjecture states that this should hold for all $S$12. A second conjecture asserts that for each fixed $S$13,

$S$14

The data support this for $S$15, while the column $S$16 does not display monotone growth in the same range (Chappelon et al., 20 Jul 2025).

4. Leaves, infinite chains, and large-scale shape

A leaf in the classical tree is a numerical semigroup with no descendants, equivalently one for which no minimal generator exceeds the Frobenius number. Leaves occur in all genera, but their distribution is highly constrained. Hyperelliptic symmetric semigroups $S$17 form an infinite chain and are not leaves, whereas non-hyperelliptic symmetric semigroups are leaves. Non-symmetric leaves also occur; one example is

$S$18

which has genus $S$19, type $S$20, gaps $S$21, and $S$22 (Chappelon et al., 20 Jul 2025).

For fixed genus $S$23, let $S$24 be the number of leaves of genus $S$25 and type $S$26. Computations up to $S$27 support the conjecture that the sequence

$S$28

is unimodal. The same experiments suggest an upper envelope for the type of a leaf: if $S$29 with $S$30, then a leaf $S$31 appears to satisfy

$S$32

The proportion of leaves also seems to rise slowly with genus: the ratio $S$33 increases from approximately $S$34 at $S$35 to approximately $S$36 at $S$37 (Chappelon et al., 20 Jul 2025).

A different large-scale phenomenon is the existence of infinite chains. A numerical semigroup belongs to an infinite chain if and only if it has infinitely many descendants. The decisive criterion uses the positive left elements $S$38: if

$S$39

then the semigroup lies in an infinite chain exactly when $S$40. If $S$41 is composite, it lies in infinitely many infinite chains; if $S$42 is prime, the number of infinite chains is given by an explicit descendant count in the quotient-by-$S$43 construction described in the literature (Rosas-Ribeiro et al., 2023).

Despite their structural necessity, infinite chains are asymptotically negligible. For each genus $S$44, there are more semigroups of that genus not belonging to infinite chains than semigroups belonging to them (Rosas-Ribeiro et al., 2023). More strongly, if $S$45 denotes the set of genus-$S$46 semigroups, then the proportion of semigroups in $S$47 that belong to infinite chains tends to $S$48 as $S$49 (Bras-Amorós et al., 2024). This removes a persistent intuition that infinite chains might control the large-scale combinatorics of the tree.

The asymptotic picture is sharpened by degree statistics. If $S$50 denotes the limiting number of children of a randomly chosen large-genus semigroup, then

$S$51

so the limiting degree distribution is geometric with mean $S$52 and variance $S$53 (O'Dorney, 2024). A plausible implication is that the tree’s exponential growth is carried by a broad population of low-degree nodes rather than by a sparse family of highly branching vertices.

5. Computational representations and traversal methods

The numerical semigroup tree has also been a laboratory for algorithm design. One standard strategy is breadth-first traversal from $S$54, generating every child by removing each minimal generator greater than the Frobenius and then computing invariants such as $S$55, $S$56, $S$57, and $S$58 by Apéry-set or pseudo-Frobenius routines. In the type-representation work, this was implemented in GAP 4.14.0 using the numericalsgps package, producing complete tables for $S$59 and $S$60 up to $S$61 (Chappelon et al., 20 Jul 2025).

A second line of work fixes genus and Frobenius number and decomposes $S$62 into equivalence classes rooted at elementary semigroups. The map

$S$63

induces classes, each with a rooted tree structure, and in Kunz coordinates the child operation becomes a bit flip

$S$64

subject to explicit conditions (Blanco et al., 2011).

Fromentin and Hivert introduced a representation by decomposition numbers

$S$65

which makes membership and irreducibility tests local and supports a DFS using contiguous byte arrays and SIMD updates. This implementation produced the counts $S$66 up to genus $S$67 and confirmed Wilf’s conjecture for genus at most $S$68 (Fromentin et al., 2013).

Later work optimized the propagation of right generators and seeds. The RGD algorithm encodes the right generators above the Frobenius by a binary string and updates a child by a local rule whose only nontrivial test is whether $S$69 is primitive in the child (Bras-Amorós et al., 2019). The seeds approach instead broadens generators to order-$S$70 seeds and updates descendants by bitstream operations on gaps and seeds. The revisited seeds algorithm combines these bitwise operations with structural results on great-grandchildren and semigroups with at most three left elements; it was used to prove that there are no Eliahou semigroups of genus $S$71, hence Wilf’s conjecture holds up to genus $S$72, and to find three Eliahou semigroups of genus $S$73 (Bras-Amorós, 2023).

An additional refinement is the unleaved tree, in which branches that cannot reach a target genus are pruned using the gcd of the left elements and the shrinking semigroup generated by those left elements divided by their gcd. In this encoding, if $S$74 and $S$75, the whole subtree can be cut. This method yielded

$S$76

(Bras-Amorós, 18 Mar 2025).

6. Variants of the numerical semigroup tree

The phrase “numerical semigroup tree” is not confined to the classical genus-graded tree. Several related rooted structures occur in the literature.

One important variant is the tree of irreducible numerical semigroups with fixed Frobenius number $S$77. Its root is the unique irreducible semigroup $S$78 whose minimal generators are all greater than $S$79, and the children are obtained by the swap

$S$80

for minimal generators $S$81 satisfying explicit arithmetic conditions. In Kunz coordinates with respect to $S$82, this becomes a swap of bits at positions $S$83 and $S$84 in a $S$85–$S$86 vector of length $S$87 (Blanco et al., 2011).

Another family consists of fixed-genus trees built from transforms that preserve genus. The ordinarization transform

$S$88

organizes the semigroups of genus $S$89 into a rooted tree $S$90 with root

$S$91

Its depth parameter is the ordinarization number, equal to the number of nonzero nongaps not exceeding $S$92. The maximum depth is $S$93, attained uniquely by $S$94, and recent work proves that for fixed ordinarization number $S$95, the counting function in genus is eventually quasipolynomial of degree $S$96 [(Bras-Amorós, 2012); (Cyrusian et al., 11 Jun 2025)].

The quasi-ordinarization transform replaces the multiplicity by the sub-Frobenius rather than by the Frobenius. This produces, for fixed genus $S$97, a forest rooted at all quasi-ordinary semigroups of genus $S$98 together with the ordinary one. The quasi-ordinarization number is

$S$99

and for large depths the number of genus-$\mathbb N_0$00 semigroups with quasi-ordinarization number $\mathbb N_0$01 is controlled by $\mathbb N_0$02-closed sets over semigroups of smaller genus (Bras-Amorós et al., 2020).

A further variant arises from arithmetic varieties. If $\mathbb N_0$03 is an arithmetic variety, one defines a rooted tree $\mathbb N_0$04 whose vertices are the semigroups in $\mathbb N_0$05, with root $\mathbb N_0$06, and with an edge $\mathbb N_0$07 whenever $\mathbb N_0$08. This tree is not locally finite, because $\mathbb N_0$09 is infinite for every non-root $\mathbb N_0$10, but it becomes finite after imposing a Frobenius bound (Branco et al., 2023).

These variants show that the numerical semigroup tree is best understood as a family of related rooted organizations rather than a single object. The classical tree remains the universal ambient structure for genus growth, degree asymptotics, and descendant algorithms, but fixed-invariant and transformed trees isolate finer phenomena—type stabilization, ordinarization depth, irreducibility, doubling, or conductor-preserving flows—that are less visible in the unrefined genus-graded picture (Chappelon et al., 20 Jul 2025).