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I have been adding some material to matroid. I haven’t gotten around to defining oriented matroid yet (and of course there’s much besides to add).
Just a boring typesetting issue: I have changed
+-- {: .num_propoposition}
to
+-- {: .num_prop}
Unfortunately the first one does not come out as expected.
On the recent addition to matroid: graph-theorists reading “embedded subgraph” in this context are likely to find this malapropistic to the point of being wrong: “embedded graph” or “embedded graph” are usually reserved to discussions of, well, embedded graphs, i.e., discussions where there is some surface in the background, and some embedding of the abstract graph into the surface. The usual definition of “graphic matroid” does not need this. It would improve the paragraph to change “embedded subgraph” just to “subgraph”.
A more serious issue is defining a graphic matroid on the vertex set: while this can be done (somehow), by far the most usual definition of graphic matroids has the edge set as the ground, see e.g. Oxley, 2nd ed, p. 11.
Moreover, “it turns out” reads as if there is at least something to check, while in fact it is immediate from the usual definitions that dualization is involutive, and involutive with the isomorphism being a strict equality. I have much appreciation for the idea of avoiding equalities whenever possible, and it might be good to do so in this case, too, but just would like to remark that in matroid theory as it exists at the moment, the dual of the dual being the matroid itself is both strict and immediate from the definitions.
Not knowing where you are heading, I will not change it for the time being.
Peter wrote:
It would improve the paragraph to change “embedded subgraph” just to “subgraph”.
Todd, perhaps you meant “induced subgraph”?
I did mean “induced subgraph”, yes.
The word “embedded”, which you rightly point out would convey the wrong idea to graph theorists: I had in mind an analogy with (full) embeddings in category theory. But let’s change that to avoid misunderstanding. But not to just “subgraph” which might mean that not all the edges between vertices in the subgraph are actually in the subgraph.
Please feel free to make appropriate changes to the description of graphic matroid and dual matroid. I was writing in a hurry there; sorry.
By the way, Peter: any expertise you can bring to bear on graph theory matters is very appreciated! Thanks again.
@Todd, I see that last year you indicated that you had possibly done some original research on categories of matroids. Are there any updates on this?
I thought for a while that geometries (in the sense of section 4 of the page matroid) were naturally a reflective subcategory of pregeometries with reflector given by geometrification (delete the closure of $\emptyset$ and quotient by $x \sim y \iff \operatorname{cl}(x) = \operatorname{cl}(y)$). But if you use closure-preserving maps as morphisms, there doesn’t seem to be a natural way to define a counit…
Okay, the mistakes kindly pointed out by Peter have been fixed (I believe).
(J)esse: no, alas, there are no updates on this. My main interest in working on matroid at that time was more in view of model theory than anything else. But with luck I’ll get myself re-interested. :-)
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