We define a subclass of separated graphs, the class of adaptable separated graphs, and study their associated monoids. We show that these monoids are primely generated conical refinement monoids, and we explicitly determine their associated I-systems. We also show that any finitely generated conical refinement monoid can be represented as the monoid of an adaptable separated graph. These results provide the first step toward an affirmative answer to the Realization Problem for von Neumann regular rings, in the finitely generated case.

中文翻译：

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细化幺半群和自适应分离图

我们定义了分离图的子类，自适应分离图的类，并研究了它们相关的幺半群。我们表明这些幺半群是主要生成的圆锥细化幺半群，并且我们明确地确定了它们相关的 I 系统。我们还表明，任何有限生成的圆锥细化幺半群都可以表示为自适应分离图的幺半群。在有限生成的情况下，这些结果为冯诺依曼规则环的实现问题提供了肯定答案的第一步。