2024/07/05 by M. A. Moreno-Frías, J. C. Rosales · 1 citation
Computer Science · Mathematics · #Commutative Algebra and Its Applications #Graph theory and applications #Polynomial and algebraic computation
paper · pdf · doi:10.1007/s10801-024-01342-x
openalex publication_date 2024/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/24
Abstract Denote by \mathrm m(S) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> the multiplicity of a numerical semigroup S . A covariety is a nonempty family \mathscr C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> of numerical semigroups that fulfils the following conditions: there is the minimum of \mathscr C, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> the intersection of two elements of \mathscr C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> is again an element of \mathscr C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> and S\backslash \\mathrm m(S)\∈ \mathscr C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo></mml:mo> <mml:mo></mml:mo> <mml:mi>m</mml:mi> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> <mml:mo></mml:mo> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> for all S∈ \mathscr C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> such that S≠ min (\mathscr C). <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>≠</mml:mo> <mml:mo>min</mml:mo> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> In this work we describe an algorithmic procedure to compute all the elements of \mathscr C. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> We prove that there exists the smallest element of \mathscr C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> containing a set of positive integers. We show that \mathscr A(F)=\S| S \hbox is a numerical semigroup with Frobenius number F\ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo></mml:mo> <mml:mi>S</mml:mi> <mml:mo>∣</mml:mo> <mml:mi>S</mml:mi> <mml:mspace/> <mml:mtext>is a numerical semigroup with Frobenius number</mml:mtext> <mml:mspace/> <mml:mi>F</mml:mi> <mml:mo></mml:mo> </mml:mrow> </mml:math> is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.