Realization of monoids with countable sum

For every infinite cardinal number [Formula: see text], [Formula: see text]-monoids and their realization have recently been introduced and studied by Nazemian and Smertnig. It turned out that, a [Formula: see text]-monoid [Formula: see text] has a realization to a ring [Formula: see text] if there exists an element [Formula: see text] such that [Formula: see text] is [Formula: see text]-braided over [Formula: see text], and [Formula: see text], as [Formula: see text]-monoid, has a realization to [Formula: see text]. Furthermore, [Formula: see text] has a realization to hereditary rings if there exists an element [Formula: see text] such that [Formula: see text] is braided over [Formula: see text]. These prompt an investigation into when [Formula: see text]-monoids have realizations. In this paper, we discuss the realization of [Formula: see text]-monoids and provide a complete characterization for the realization of two-generated ones in hereditary Von Neumann regular rings.