Numerical Semigroups with a Given Frobenius Number and Some Fixed Gaps

María A. Moreno-Frías () and José Carlos Rosales
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María A. 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 11, 1-14

Abstract: If P is a nonempty finite subset of positive integers, then A ( P ) = { S ∣ S i s a n u m e r i c a l s e m i g r o u p , S ∩ P = ∅ a n d max ( P ) i s t h e F r o b e n i u s n u m b e r o f S } . In this work, we prove that A ( P ) is a covariety; therefore, we can arrange the elements of A ( P ) in the form of a tree. This fact allows us to present several algorithms, including one that calculates all the elements of A ( P ) , another that obtains its maximal elements (with respect to the set inclusion order) and one more that computes the elements of A ( P ) that cannot be expressed as an intersection of two elements of A ( P ) , that properly contain it.

Keywords: Frobenius number; gap; multiplicity; algorithm; covariety; irreducible element; R variety (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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