We say that a numerical semigroup is
$d$
-squashed if it can be written in the form
$$S\,=\,\frac{1}{N}\langle {{a}_{1}},\,.\,.\,.\,,\,{{a}_{d}}\rangle \,\cap \,\mathbb{Z}$$
for
$N$
,
${{a}_{1}}\,,\,.\,.\,.\,,\,{{a}_{d}}$
positive integers with
$\gcd \left( {{a}_{1}},\,.\,.\,.\,,{{a}_{d}} \right)\,=\,1$
. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular.
Recent works by Rosales et al. give a concrete example of a numerical semigroup that cannot be written as an intersection of 2-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of 2-squashed semigroups. We also will prove the same result for 3-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of
$d$
-squashed semigroups for any fixed
$d$
, and we prove some partial results towards this conjecture.