论文标题
本地最终的广泛类别，半环和连接对象的分解
Locally-finite extensive categories, their semi-rings, and decomposition to connected objects
论文作者
论文摘要
令$ \ Mathcal C $为有限图的类别。 lovàsz表明，$ \ MATHCAL C $的同构类别的半环（以相关为单位，以及作为乘法的乘积）嵌入到自然数的半环的直接产物中。我们的目的是将此结果概括为其他类别。为此，一个至关重要的属性是每个对象分解为连接对象的有限共同点。我们表明，当地的广泛类别满足了这种情况。相反，将任何对象分解为连接对象的有限共同点的类别被证明是广泛的。分解原来是独一无二的。使用这些结果，我们提供了一些足够的条件，即局部有限类别的同构类别的半环（环）嵌入自然数的直接乘积（分别是整数）。这种类别的环的结构是伯恩赛德环和绿色环的一种最原始的形式。
Let $\mathcal C$ be the category of finite graphs. Lovàsz shows that the semi-ring of isomorphism classes of $\mathcal C$ (with coproduct as sum, and product as multiplication) is embedded into the direct product of the semi-ring of natural numbers. Our aim is to generalize this result to other categories. For this, one crucial property is that every object decomposes to a finite coproduct of connected objects. We show that a locally-finite extensive category satisfies this condition. Conversely, a category where any object is decomposed into a finite coproduct of connected objects is shown to be extensive. The decomposition turns out to be unique. Using these results, we give some sufficient conditions that the semi-ring (the ring) of isomorphism classes of a locally finite category embeds to the direct product of natural numbers (integers, respectively). Such a construction of rings from a category is a most primitive form of Burnside rings and Grothendieck rings.
