Modular retractions of numerical semigroups

Let S be a numerical semigroup, let m be a nonzero element of S , and let a be a nonnegative integer. We denote (where as mod m is the remainder of the division of as by m ). In this paper we characterize the pairs ( a , m ) such that 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 1 mod m ,…, n p − an p mod m 〉. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type.