On the Enumeration of the Set of Numerical Semigroups with Fixed Frobenius Number and Fixed Number of Second Kind Gaps

We study how certain invariants of numerical semigroups relate to the number of second kind gaps. Furthermore, given two fixed non-negative integers F and k, we provide an algorithm to compute all the numerical semigroups whose Frobenius number is F and which have exactly k second kind gaps.

Modular Frobenius pseudo-varieties

If $$m \in {\mathbb {N}} \setminus \{0,1\}$$ m ∈ N \ { 0 , 1 } and A is a finite subset of $$\bigcup _{k \in {\mathbb {N}} \setminus \{0,1\}} \{1,\ldots ,m-1\}^k$$ ⋃ k ∈ N \ { 0 , 1 } { 1 , … , m - 1 } k , then we denote by $$\begin{aligned} {\mathscr {C}}(m,A) =&\{ S\in {\mathscr {S}}_m \mid s_1+\cdots +s_k-m \in S \text { if } (s_1,\ldots ,s_k)\in S^k \text { and } \\ {}&\qquad (s_1 \bmod m, \ldots , s_k \bmod m)\in A \}. \end{aligned}$$ C ( m , A ) = { S ∈ S m ∣ s 1 + ⋯ + s k - m ∈ S if ( s 1 , … , s k ) ∈ S k and ( s 1 mod m , … , s k mod m ) ∈ A } . In this work we prove that $${\mathscr {C}}(m,A)$$ C ( m , A ) is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to $${\mathscr {C}}(m,A)$$ C ( m , A ) and to compute all the elements of $${\mathscr {C}}(m,A)$$ C ( m , A ) with a fixed genus. Moreover, we introduce and study three families of numerical semigroups, called of second-level, thin and strong, and corresponding to $${\mathscr {C}}(m,A)$$ C ( m , A ) when $$A=\{1,\ldots ,m-1\}^3$$ A = { 1 , … , m - 1 } 3 , $$A=\{(1,1),\ldots ,(m-1,m-1)\}$$ A = { ( 1 , 1 ) , … , ( m - 1 , m - 1 ) } , and $$A=\{1,\ldots ,m-1\}^2 \setminus \{(1,1),\ldots ,(m-1,m-1)\}$$ A = { 1 , … , m - 1 } 2 \ { ( 1 , 1 ) , … , ( m - 1 , m - 1 ) } , respectively.

Infinite sequences of almost symmetric non-Weierstrass numerical semigroups

Let u be any positive integer. We construct infinite sequences of almost symmetric non-Weierstrass numerical semigroups whose conductors are the genera double minus $$2u-1$$ 2 u - 1 . Moreover, let v be any non-negative integer. We give an infinite number of non-Weierstrass numerical semigroups whose conductors are the genera double minus 2v.

Numerical semigroups bounded by the translation of a plane monoid

Let $$\mathbb {N}$$ N be the set of nonnegative integer numbers. A plane monoid is a submonoid of $$(\mathbb {N}^2,+)$$ ( N 2 , + ) . Let M be a plane monoid and $$p,q\in \mathbb {N}$$ p , q ∈ N . We will say that an integer number n is M(p, q)-bounded if there is $$(a,b)\in M$$ ( a , b ) ∈ M such that $$a+p\le n \le b-q$$ a + p ≤ n ≤ b - q . We will denote by $${\mathrm A}(M(p,q))=\{n\in \mathbb {N}\mid n \text { is } M(p,q)\text {-bounded}\}.$$ A ( M ( p , q ) ) = { n ∈ N ∣ n is M ( p , q ) -bounded } . An $$\mathcal {A}(p,q)$$ A ( p , q ) -semigroup is a numerical semigroup S such that $$S= {\mathrm A}(M(p,q))\cup \{0\}$$ S = A ( M ( p , q ) ) ∪ { 0 } for some plane monoid M. In this work we will study these kinds of numerical semigroups.

Numerical semigroups, polyhedra, and posets II: locating certain families of semigroups

Several recent papers have examined a rational polyhedron Pm whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing m. A combinatorial description of the faces of Pm was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of Pm containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of Pm . In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of Pm .