Realization of monoids with countable sum

Abstract: 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.