  For every infinite cardinal number $\kappa$, $\kappa$-monoids and their
realization have recently been introduced and studied by Nazemian and Smertnig.
A $\kappa$-monoid $H$ has a realization to a ring $R$ if there exists an
element $x \in H$ such that $H$ is $\aleph_1 ^{-}$-braided over
$\text{add}(\aleph_0 x)$, and $\text{add}(\aleph_0 x)$, as $\aleph_0$-monoid,
has a realization to $R$. Furthermore, $H$ has a realization to hereditary
rings if there exists an element $x \in H$ such that $H$ is braided over
$\text{add}(x)$. These prompt an investigation into when $\aleph_0$-monoids
have realizations. In this paper, we discuss the realization of
$\aleph_0$-monoids and provide a complete characterization for the realization
of two-generated ones in hereditary Von Neumann regular rings.
