Union of Sets of Lengths of Numerical Semigroups

J. I. García-García, D. Marín-Aragón and A. Vigneron-Tenorio
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J. I. García-García: Departamento de Matemáticas/INDESS (Instituto Universitario para el Desarrollo Social Sostenible), Universidad de Cádiz, E-11510 Puerto Real, Spain
D. Marín-Aragón: Departamento de Matemáticas, Universidad de Cádiz, E-11510 Puerto Real, Spain
A. Vigneron-Tenorio: Departamento de Matemáticas/INDESS (Instituto Universitario para el Desarrollo Social Sostenible), Universidad de Cádiz, E-11406 Jerez de la Frontera, Spain

Mathematics, 2020, vol. 8, issue 10, 1-8

Abstract: Let S = 〈 a 1 , … , a p 〉 be a numerical semigroup, let s ∈ S and let Z ( s ) be its set of factorizations. The set of lengths is denoted by L ( s ) = { L ( x 1 , ? , x p ) ? ( x 1 , ? , x p ) ∈ Z ( s ) } , where L ( x 1 , ? , x p ) = x 1 + ? + x p . The following sets can then be defined: W ( n ) = { s ∈ S ? ∃ x ∈ Z ( s ) such that L ( x ) = n } , ν ( n ) = ? s ∈ W ( n ) L ( s ) = { l 1 < l 2 < ? < l r } and Δ ν ( n ) = { l 2 − l 1 , … , l r − l r − 1 } . In this paper, we prove that the function Δ ν : N → P ( N ) is almost periodic with period lcm ( a 1 , a p ) .

Keywords: delta-set; non-unique factorization; numerical monoid; numerical semigroup (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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