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If S is a numerical semigroup, denote by g (S) the genus of S. A numerical semigroup T is an I (S) -semigroup if T ∖ { 0 } is an ideal of S. If k ∈ ℕ , then we denote by i (S , k) the number of I (S) -semigroups with genus g (S) + k. In this work, we conjecture that i (S , a) ≤ i (S , b) if a ≤ b , and we show that there is a term from which this sequence becomes stationary. That is, there exists k S ∈ ℕ such that i (S , k S) = i (S , k S + h) for all h ∈ ℕ. Moreover, we prove that the conjecture is true for ordinary numerical semigroups, that is, numerical semigroups which the form { 0 , m , → } for some positive integer. Additionally, we calculate the term from which the sequence becomes stationary. [ABSTRACT FROM AUTHOR]

We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP [The GAP Group, GAP — Groups, Algorithms and Programming, Version 4.8.6; 2016, https://www.gap-system.org ] package NumericalSgps [M. Delgado and P. A. García-Sánchez and J. Morais, "numericalsgps": A GAP package on numerical semigroups, https://github.com/gap-packages/numericalsgps ]. [ABSTRACT FROM AUTHOR]

We present procedures to calculate the set of Arf numerical semigroups with given genus, given conductor and given genus and conductor. We characterize the Kunz coordinates of an Arf numerical semigroup. We also describe Arf numerical semigroups with fixed Frobenius number and multiplicity up to 7. [ABSTRACT FROM AUTHOR]

We characterize affine semigroups having one Betti element and we compute some relevant non-unique factorization invariants for these semigroups. As an example, we particularize our description to numerical semigroups. [ABSTRACT FROM AUTHOR]

We study the relationship between numerical semigroups and periodic subadditive functions. [ABSTRACT FROM AUTHOR]