One half of almost symmetric numerical semigroups

Let $S,T$ be two numerical semigroups. We study when $S$ is one half of $T$, with $T$ almost symmetric. If we assume that the type of $T$, $t(T)$, is odd, then for any $S$ there exist infinitely many such $T$ and we prove that $1 \leq t(T) \leq 2t(S)+1$\mathrm{.}Ontheotherhand,if$t(T)$iseven,thereexistssuch$T$ifandonlyif$S$isalmostsymmetricanddifferentfrom$$\mathbb{N}$$; in this case the type of $S$ is the number of even pseudo-Frobenius numbers of $T$. Moreover, we construct these families of semigroups using the numerical duplication with respect to a relative ideal.