Realization of monoids with countable sum

For every infinite cardinal number, -monoids and their realization have recently been introduced and studied by Nazemian and Smertnig. A -monoid H has a realization to a ring R if there exists an element x H such that H is ₁ ^--braided over add (₀ x), and add (₀ x), as ₀-monoid, has a realization to R. Furthermore, H has a realization to hereditary rings if there exists an element x H such that H is braided over add (x). These prompt an investigation into when ₀-monoids have realizations. In this paper, we discuss the realization of ₀-monoids and provide a complete characterization for the realization of two-generated ones in hereditary Von Neumann regular rings.