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I had posted this on Stackexchange because I don't believe this is a particlarly difficult question, but there were no answers, so I'm posting it on here now.

I just had a quick question about the definition of a loop for an oriented matroid. This issue stemmed from trying to determine what the sign would be associated to a loop in a digraph when applying an oriented matroid structure onto it, and I soon ran into some trouble.

In the textbook Oriented Matroids by Bjorner et al., given an oriented matroid $\mathcal{M} = (E, \mathcal{C})$, where $\mathcal{C}$ are the signed circuits, a loop is defined to be an element $e \in E$ such that $(\{e\}, \varnothing) \in \mathcal{C}$, where the notation $X = (X^+, X^-)$ represents that $X^+$ is the positive part of a signed set $X$, and $X^-$ is the negative part. Because $\mathcal{C} = -\mathcal{C}$ by the signed circuit axioms, a loop $\{e\}$ will have signs $(+)$ and $(-)$ associated with it.

However, in these notes, in Definition 1.4, an element $e$ is called a loop if, for any signed set containing $e$, the sign associated to $e$ is $0$.

To me, it seems like these definitions are contradictory, and the more I try to find resources online to clear it up, the more I see some sources giving the former definition, and then others giving the latter. I would sincerely appreciate some clarity on the matter!

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    $\begingroup$ This is probably not that helpful of a comment, but whatever a loop is in an oriented matroid, it ought to be a loop of the underlying unoriented matroid. $\endgroup$ Commented Oct 15, 2024 at 1:30
  • $\begingroup$ Thank you for the answer! What seems particularly odd is that, at least from what I can tell, both definitions provide loops in the underlying matroid! But I think this just follows from the fact that $\{e\}$ is in the signed circuits, and so will be in the underlying circuit. $\endgroup$ Commented Oct 15, 2024 at 19:51

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