Union of Sets of Lengths of Numerical Semigroups

Abstract

Let S = < a(1), ... , a(p)> be a numerical semigroup, let s is an element of S and let Z(s) be its set of factorizations. The set of lengths is denoted by L(s) = {L(x(1), ... , x(p)) vertical bar (x(1), ... , x(p)) is an element of Z(s)}, where L(x(1), ... , x(p)) = x(1) + ... + x(p). The following sets can then be defined: W(n) = {s is an element of S vertical bar there exists x is an element of Z(s) such that L(x) = n}, nu(n) = boolean OR(s is an element of W(n)) L(s) = {l(1) < l(2) < ... < l(r)} and Delta nu(n) = {l(2) l(1), ... , l(r) - l(r-1)}. In this paper, we prove that the function Delta nu : N -> P(N) is almost periodic with period lcm(a(1), a(p)).