Modular retractions of numerical semigroups

Abstract

Let S be a numerical semigroup, let m be a nonzero element of S, and let a be a nonnegative integer. We denote ${\rm R}(S,a,m) = \{ s-as \bmod m \mid s \in S \}$ (where asmodm is the remainder of the division of as by m). In this paper we characterize the pairs (a,m) such that ${\rm R}(S,a,m)$ is a numerical semigroup. In this way, if we have a pair (a,m) with such characteristics, then we can reduce the problem of computing the genus of S=〈n 1,…,n p 〉 to computing the genus of a “smaller” numerical semigroup 〈n 1−an 1modm,…,n p −an p modm〉. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type.