AbstractIn this paper we present and study the ideal duplication, a new construction within the class of the relative ideals of a numerical semigroup S, that, under specific assumptions, produces a relative ideal of the numerical duplication $$S\bowtie ^b E$$
S
⋈
b
E
. We prove that every relative ideal of the numerical duplication can be uniquely written as the ideal duplication of two relative ideals of S; this allows us to better understand how the basic operations of the class of the relative ideals of $$S\bowtie ^b E$$
S
⋈
b
E
work. In particular, we characterize the ideals E such that $$S\bowtie ^b E$$
S
⋈
b
E
is nearly Gorenstein.