Combinatorial quotients

Consider a combinatorial class (in the sense of Flajolet and Sedgewick) or combinatorial species (in the sense of Bergeron, Labelle, and Leroux). There are certain generating functions associated with combinatorial classes/species (I'm conflating the two for the moment because the difference between the two theories aren't important here), and the operations of sum and product have nice "combinatorial" interpretations -- addition of generating functions corresponds to disjoint unions of classes/species and multiplication of generating functions what I'll call here "combinatorial product", i. e. for two species F and G, an FG-structure on the set [n] is a pair which consists of an F-structure on some subset S of [n] and a G-structure on [n]-S. Then the generating function of the FG-structures is the product of the generating function of the F-structures and that of the G-structures.

Other natural operations on functions have combinatorial interpretations: composition of functions, taking exponentials or logarithms, differentiation, integration, etc. Even subtraction has an interpretation, although one has to be careful: this is the "virtual species". (A virtual species is a pair of ordinary species (F, G), written as F-G, where (F, G) = (H, K) if F + K = G + H. Replacing species with integers and addition with multiplication, this looks like the construction of the rational numbers from the integers.)

But is there a natural quotient operation? If there is, I can't find a mention of it in Flajolet and Sedgewick, and Marko Petkovsek (who I was just talking about this with) tells me it's not in Bergeron et al. either; this is one of the sources he's using for our current topics course in combinatorics. Let F be a species with 1 structure on the set of size 0, and let F_n be its restriction to size n. Let F₊ be its restriction to all positive sizes. Then

where 1 is the multiplicative identity species (hence the name); it has one structure on a set of size 0. We can of course write this as

and we collect the "positive" and "negative" terms to get

So the inverse of the species F exists, and it's a virtual species in which the positive part is even-length tuples of nonempty F-structures, and the negative part is odd-length tuples of nonempty F-structures. This doesn't quite seem like a "combinatorially natural" operation -- but on the other hand I was able to describe it in a single sentence, so it seems plausible that it could be a natural thing to occur somewhere.

Notice that this also proves that if

and F(z)G(z) = 1, f₀ = 1, then all the g_n are integers, since the generating function G(z) actually counts something! (Well, a virtual something -- but that's close enough. It won't surprise you to learn that for two species H and K, the generating function corresponding to the virtual species H-K is H(z) - K(z). G here is a virtual species, so the g_n are (possibly negative) integers. Of course, there are other ways to prove this, but I think this is a cute "combinatorial proof".