The Covariety of Saturated Numerical Semigroups with Fixed Frobenius Number
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MDPI
Materia
Numerical semigroup Covariety Frobenius number
Date
2024-06-03Referencia bibliográfica
Rosales, J.C.; Moreno-Frías, M.Á. The Covariety of Saturated Numerical Semigroups with Fixed Frobenius Number. Foundations 2024, 4, 249–262. https://doi.org/10.3390/foundations4020016
Sponsorship
Proyecto de Excelencia de la Junta de Andalucía Grant Number ProyExcel_00868; Junta de Andalucía Grant Number FQM-343; Junta de Andalucía Grant Number FQM-298; Proyecto de investigación del Plan Propio—UCA 2022-2023 (PR2022-004)Abstract
In this work, we show that if F is a positive integer, then Sat(F) = {S | S is a saturated numerical
semigroup with Frobenius number F} is a covariety. As a consequence, we present two
algorithms: one that computes Sat(F), and another which computes all the elements of Sat(F) with
a fixed genus. If X ⊆ S\Δ(F) for some S ∈ Sat(F), then we see that there exists the least element
of Sat(F) containing X. This element is denoted by Sat(F)[X]. If S ∈ Sat(F), then we define the
Sat(F)-rank of S as the minimum of {cardinality(X) | S = Sat(F)[X]}. In this paper, we present an
algorithm to compute all the elements of Sat(F) with a given Sat(F)-rank.