Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an $E^*$-unitary inverse semigroup. As a consequence, the tight groupoid of this semigroup is a Hausdorff \'etale groupoid. We show that this groupoid is always amenable, and that the type semigroups of groupoids obtained from adaptable separated graphs in this way include all finitely generated conical refinement monoids. The first three named authors will utilize this construction in forthcoming work to solve the Realization Problem for von Neumann regular rings, in the finitely generated case.

中文翻译：

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自适应分离图的群群及其类型半群

给定一个自适应分离图，我们构造一个关联的 groupoid 并探索它的类型半群。具体来说，我们首先将一个对应的半群附加到每个自适应分离图上，我们证明它是一个 $E^*$-酉逆半群。因此，这个半群的紧群群是 Hausdorff \'etale groupoid。我们证明了这个 groupoid 总是适合的，并且以这种方式从自适应分离图中获得的 groupoids 的类型半群包括所有有限生成的圆锥细化幺半群。在有限生成的情况下，前三位作者将在即将开展的工作中利用这种结构来解决冯诺依曼规则环的实现问题。