I do have a few examples where nodes and edges have labels that form a semiring, and so far these semirings were idempotent, to match the a + a = a law in the algebra (a natural definition for labelled graph overlay is [x]a + [y]a = [x y]a).

Your suggestion to consider labels forming a group is interesting. Indeed, this will give negation, even though we’ll then lose the idempotence of graph overlay. I’ll look into it!

]]>The intuition I have, with multisets, is fibres of species are like multisets. Ostensibly they’re sets, but since the sum is tagged, they are more like multisets: L+L has two of each lists at each size. And sum is fibre-wise. It’s kind of where sum and product differ: products modify what « shapes » are available at each fibre, for sums, the « shape » of object is opaque, so we might as well take blurry glasses and see them as atoms. So, from this point of view, sums are pretty much just fibre-wise multiset sums (and subtraction, fibre-wise « virtual » multiset subtraction).

Regarding List-Bags: that’s indeed what I meant. And I realise that I was abusing the notation a bit (we could say that it’s all the lists except one per size, then every choice of the one missing list yields a syntactically different, but equivalent, species, but that’s rather boring…).

]]>The underlying solutions are basically the same, but your interpretation of this as a boolean algebra produced a very elegant implementation. Beautiful! ]]>

For example:

Using F T under xor gets you sum where a graph is its own inverse. Weighting by Z gives you Multigraphs that can have negative number of edges. Wightin by R+ (and using multiplication) you get something really weird.

But I think your addition is closer to an meet/union then a disjoint union so maybe look at lattices? Or at the min,+ rig?

You have probably considered all this.

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