Numerical semigroups problem list

On the genus of a quotient of a numerical semigroup

“…This result generalizes the formulas for g( a, b /2) (when gcd(a, b) = 1) given in [4] to formulas for g( a, b /d) for all d ≥ 2. The latter result answers an open problem listed in [3].…”

“…One motivation for studying quotients of numerical semigroups comes from the study of proportionally modular Diophantine inequalities, which are Diophantine inequalities of the form ax (mod b) ≤ cx for some fixed positive integers a, b, c. It turns out that the set of nonnegative integer solutions to a proportionally modular Diophantine inequality form a numerical semigroup; a numerical semigroup obtained in this way is called a proportionally modular numerical semigroup. Robles-Pérez and Rosales [12] have shown that a numerical semigroup is proportionally modular if and only if it is of the form a, a + 1 /d for some positive integers a and d. Furthermore, Delgado, García-Sánchez, and Rosales [3] have remarked that there is no known example of a numerical semigroup that is not of the form a, b, c /d. It is natural to study quotients of the special numerical semigroups whose invariants are already well understood.…”

“…, a n }. It is well known that every numerical semigroup is of this form [3,Theorem 2.7]. That is, every numerical semigroup is generated by a finite set of positive integers.…”

Categories of two-colored pair partitions part I: categories indexed by cyclic groups

“…Remark 7.18. Submonoids M of (N 0 , +) with finite N 0 M , also known as numerical semigroups, are interesting as such (see [DGR13] for a survey on open problems in this area). Given a finitely generated bracket pattern category W, the set N 0 ⋃ A(W) is the largest numerical semigroup disjoint from ⋃ W. (In fact, one can show ⋃ A(W) = ⋃ W.) Its gap set is ⋃ A(W), its genus A(⋃ W) and its Frobenius number ⋃ W .…”

An extension of the Wilf conjecture to semimodules over a numerical semigroup

“…More recently, S. Böcker and Z. Lipták [3] improved the time complexity of Nijenhuis' algorithm to O(e(Γ)a 1 ) with interesting applications to interpreting mass spectrometry peaks. Based on those improvements as well as on the huge number of examples provided by many of the above mentioned references [7,5,9,11,8], one realizes that the inequality in the Wilf conjecture 1.1 seems to be far from being tight. Having this in mind, the underlying meaning of the Wilf conjecture reveals to be the finding of an upper bound for the quotient c(Γ)/δ (Γ) and, after that, the comparison of this bound with the embedding dimension of the semigroup.…”