Talk:Equivalence of Definitions of Matroid Circuit Axioms

Condition vs Formulation

I used the term "Condition" as I didn't think Condition 4 was the formulation of axioms. --Leigh.Samphier (talk) 11:25, 2 February 2023 (UTC)

Trouble is, you are also using the word "condition" to mean the axioms themselves.

That is not the case. I'm saying that the Condition is that $\mathscr C$ satisfies the axioms, not that the Condition is the axioms. --Leigh.Samphier (talk) 12:42, 2 February 2023 (UTC)

In Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 2 we have:

"Let $\mathscr C$ satisfy conditions $(\text C 1)$, $(\text C 2)$ and $(\text C 3)$."

I still think there is plenty of room for clarification. --prime mover (talk) 20:42, 2 February 2023 (UTC)

Thanks for pointing that out, that was incorrect. If you notice any other instances like that let me know. --Leigh.Samphier (talk) 08:24, 3 February 2023 (UTC)

I wonder whether to leave formulation 4 out of this page as it is out of the scope of the equivalence of the 3 formulations. Either that, or enter 4 as another formulation. It's confusing as it stands. --prime mover (talk) 12:28, 2 February 2023 (UTC)

Each Condition is a statement about $\mathscr C$. The 4 statements about $\mathscr C$ are equivalent. Where is the confusion? --Leigh.Samphier (talk) 12:42, 2 February 2023 (UTC)

$\mathscr C$ is not specifically defined in a definition page. All you have is the three equivalent formulations of the circuit axioms, but (except, seemingly as an afterthought, as a 4th "conditioN" on this equivalence page) there isn't a definition page which explains what the object whose behaviour is specified by the circuit axioms actually is. --prime mover (talk) 12:52, 2 February 2023 (UTC)

I think the problem is that $\mathscr C$ has no other name than the set of circuits of a matroid, as far as I can tell.

The situation is similar to the difference between a synthetic basis and an analytic basis in topology. On the one hand $\mathscr C$ is derived from a matroid. On the other $\mathscr C$ derives a matroid. I can't see in the limited literature that I have where this distinction is made explicit.

I'll need to give it some more thought. --Leigh.Samphier (talk) 09:43, 3 February 2023 (UTC)

Is this what you imagined? I sepeareted the theroem into two. See User:Leigh.Samphier/ForReview/MatroidCircuitAxioms --Leigh.Samphier (talk) 09:12, 6 February 2023 (UTC)

There's a lot to hack through. It will need to wait for a time when my mind is fresh, which inevitably means a weekend. --prime mover (talk) 17:09, 6 February 2023 (UTC)