We show that every finitely generated conical refinement monoid can be represented as the monoid \(\mathcal V (R)\) of isomorphism classes of finitely generated projective modules over a von Neumann regular ring R. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph (E, C) and a field K, we build a von Neumann regular K-algebra \(Q_K(E,C)\) and show that there is a natural isomorphism between the separated graph monoid M(E, C) and the monoid \(\mathcal V (Q_K(E,C))\).

中文翻译：

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有限生成的精简半形体的实现问题

我们证明了，每个有限生成的圆锥细化等式都可以表示为von Neumann正则环R上有限生成的射影模块的同构类的等分线\（\ mathcal V（R）\）。为此，我们使用可适应的分离图提供的这些类四面体的表示。给定一个可适应的分离图（E，  C）和一个场K，我们建立了von Neumann正则K-代数\（Q_K（E，C）\）并表明在分离图半分体M（E，  C）和monoid\（\数学V（Q_K（E，C））\）。