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The collection of numerical semigroups that contain a given finite set ℓP of positive integers as part of their pseudo‑Frobenius numbers—and whose Frobenius number is the largest element of ℓP—turns out to be a covariety. For any ℓP, all semigroups in this set share the same Frobenius number, which is ∎ℓP.
A constructive tree gives every member of the covariety. Starting from the minimal semigroup ∏( C(ℓP) )={0,max(P)+1,ℕ} the algorithm adds a new generator whenever it can keep the multiplicity below the maximum. At each node the Apery set is used to compute the multiplicity, the set of pseudo‑Frobenius numbers and the stingy gaps. This method enumerates the tree level by level, producing the entire set C(ℓP). Their levels encode the genus: a semigroup at level d has genus g=max(P)−d.
| d | msg(S) | parent | t(S) | PF(S) | |
|---|---|---|---|---|---|
| S0 | 0 | ⟨12,13,…,23⟩ | – | 11 | {1,2,…,11} |
| S1 | 1 | ⟨6,13,14,15,16,17⟩ | S0 | 5 | {7,8,9,10,11} |
| S2 | 1 | ⟨7,12,13,15,16,17,18⟩ | S0 | 6 | {5,6,8,9,10,11} |
| S3 | 1 | ⟨8,12,13,14,15,17,18,19⟩ | S0 | 7 | {4,5,6,7,9,10,11} |
| S4 | 1 | ⟨10,12,13,…,19,21⟩ | S0 | 9 | {2,3,…,9,11} |
| S5 | 2 | ⟨4,13,14,15⟩ | S3 | 3 | {9,10,11} |
| S6 | 2 | ⟨5,12,13,14,16⟩ | S4 | 4 | {7,8,9,11} |
| S7 | 2 | ⟨6,7,15,16,17⟩ | S2 | 4 | {8,9,10,11} |
| S8 | 2 | ⟨6,8,13,15,17⟩ | S3 | 4 | {7,9,10,11} |
| S9 | 2 | ⟨6,10,13,14,15,17⟩ | S4 | 5 | {4,7,8,9,11} |
| S10 | 2 | ⟨7,8,12,13,17,18⟩ | S3 | 5 | {5,6,9,10,11} |
| S11 | 2 | ⟨7,10,12,13,15,16,18⟩ | S4 | 6 | {3,5,6,8,9,11} |
| S12 | 2 | ⟨8,10,12,13,14,15,17,19⟩ | S4 | 7 | {2,4,5,6,7,9,11} |
| S13 | 3 | ⟨4,10,13,15⟩ | S12 | 3 | {6,9,11} |
| S14 | 3 | ⟨5,7,13,16⟩ | S11 | 3 | {8,9,11} |
| S15 | 3 | ⟨5,8,12,14⟩ | S12 | 3 | {7,9,11} |
| S16 | 3 | ⟨6,7,8,17⟩ | S10 | 3 | {9,10,11} |
| S17 | 3 | ⟨6,7,10,15⟩ | S11 | 3 | {8,9,11} |
| S18 | 3 | ⟨6,8,10,13,15,17⟩ | S12 | 5 | {2,4,7,9,11} |
| S19 | 3 | ⟨7,8,10,12,13⟩ | S12 | 4 | {5,6,9,11} |
| S20 | 4 | ⟨4,6,13,15⟩ | S18 | 3 | {2,9,11} |
| S21 | 4 | ⟨5,7,8⟩ | S19 | 2 | {9,11} |
| S22 | 4 | ⟨6,7,8,10⟩ | S19 | 2 | {9,11} |
Because the tree is finite, C(ℓP) contains only finitely many semigroups. The authors list all elements for the example ℓP={9,11}. The table shows their multiplicities, embedding dimensions, types, minimal generating sets (restricted to numbers below 11), ranks and whether they have maximal embedding dimension or are maximal in the covariety.
| m(S) | e(S) | t(S) | 𝒞(𝒫)msg(S) | 𝒞(𝒫)rank(S) | MED | maximal | |
|---|---|---|---|---|---|---|---|
| S0 | 12 | 12 | 11 | ∅ | 0 | yes | no |
| S1 | 6 | 6 | 5 | {6} | 1 | yes | no |
| S2 | 7 | 7 | 6 | {7} | 1 | yes | no |
| S3 | 8 | 8 | 7 | {8} | 1 | yes | no |
| S4 | 10 | 10 | 9 | {10} | 1 | yes | no |
| S5 | 4 | 4 | 3 | {4} | 1 | yes | no |
| S6 | 5 | 5 | 4 | {5} | 1 | yes | no |
| S7 | 6 | 5 | 4 | {6,7} | 2 | no | no |
| S8 | 6 | 5 | 4 | {6,8} | 2 | no | no |
| S9 | 6 | 6 | 5 | {6,10} | 2 | yes | no |
| S10 | 7 | 6 | 5 | {7,8} | 2 | no | no |
| S11 | 7 | 7 | 6 | {7,10} | 2 | yes | no |
| S12 | 8 | 8 | 7 | {8,10} | 2 | yes | no |
| S13 | 4 | 4 | 3 | {4,10} | 2 | yes | no |
| S14 | 5 | 4 | 3 | {5,7} | 2 | no | no |
| S15 | 5 | 4 | 3 | {5,8} | 2 | no | no |
| S16 | 6 | 4 | 3 | {6,7,8} | 3 | no | no |
| S17 | 6 | 4 | 3 | {6,7,10} | 3 | no | no |
| S18 | 6 | 6 | 5 | {6,8,10} | 3 | yes | no |
| S19 | 7 | 5 | 4 | {7,8,10} | 3 | no | no |
| S20 | 4 | 4 | 3 | {4,6} | 2 | yes | yes |
| S21 | 5 | 3 | 2 | {5,7,8} | 3 | no | yes |
| S22 | 6 | 4 | 2 | {6,7,8,10} | 4 | no | yes |
These results give the first complete classification of numerical semigroups with a specified set of pseudo‑Frobenius numbers and provide algorithms that can be implemented with standard integer‑algebra tools.