Dilatations of numerical semigroups

Abstract

This paper is focused on numerical semigroups and presents a simple construction, that we call dilatation, which, from a starting semigroup $S$, permits to get an infinite family of semigroups which share several properties with $S$. The invariants of each semigroup $T$ of this family are given in terms of the corresponding invariants of $S$ and the Ap\'ery set and the minimal generators of $T$ are also described. We also study three properties that are close to the Gorenstein property of the associated semigroup ring: almost Gorenstein, 2-AGL, and nearly Gorenstein properties. More precisely, we prove that $S$ satisfies one of these properties if and only if each dilatation of $S$ satisfies the corresponding one.