The Set of Numerical Semigroups with Frobenius Number Belonging to a Fixed Interval

María Ángeles Moreno-Frías () and José Carlos Rosales
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María Ángeles Moreno-Frías: Department of Mathematics, Faculty of Sciences, University of Cádiz, E-11510 Cádiz, Spain
José Carlos Rosales: Department of Algebra, Faculty of Sciences, University of Granada, E-18071 Granada, Spain

Mathematics, 2025, vol. 13, issue 15, 1-15

Abstract: Let a and b be positive integers such that a < b and [ a , b ] = { x ∈ N ∣ a ≤ x ≤ b } . In this work, we will show that A ( [ a , b ] ) = { S ∣ S is a numerical semigroup whose Frobenius number belongs to [ a , b ] } and is a covariety. This fact allows us to present an algorithm which computes all the elements from A ( [ a , b ] ) . We will prove that A ( [ a , b ] , m ) = { S ∈ A ( [ a , b ] ) ∣ S has multiplicity m } and is a ratio-covariety. As a consequence, we will show an algorithm which calculates all the elements belonging to A ( [ a , b ] , m ) . Based on the above results, we will develop an interesting algorithm that calculates all numerical semigroups with a given multiplicity and complexity.

Keywords: Frobenius number; multiplicity; algorithm; covariety; ratio-covariety; complexity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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