Locally-finite extensive categories, their semi-rings, and decomposition to connected objects

Let $\mathcal C$ be the category of finite graphs. Lov\`{a}sz 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.