The covariety of numerical semigroups with fixed Frobenius number

• Published in: Journal of algebraic combinatorics Vol. 60; no. 2; pp. 555 - 568
• Main Authors: Moreno-Frías, M. A., Rosales, J. C.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2024; Springer Nature B.V
• Subjects: Children & youth; Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Lattices; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Semigroups; Covariety; Genus; Multiplicity; Numerical semigroup; Frobenius number; Rank; Algorithm
• ISSN: 0925-9899; 1572-9192; 1572-9192
• DOI: 10.1007/s10801-024-01342-x

Denote by m ( S ) the multiplicity of a numerical semigroup S . A covariety is a nonempty family C of numerical semigroups that fulfils the following conditions: there is the minimum of C , the intersection of two elements of C is again an element of C and S \ { m ( S ) } ∈ C for all S ∈ C such that S ≠ min ( C ) . In this work we describe an algorithmic procedure to compute all the elements of C . We prove that there exists the smallest element of C containing a set of positive integers. We show that A ( F ) = { S ∣ S is a numerical semigroup with Frobenius number F } 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.