Definition 1 : A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A fan is a finite set $F$ of polyhedral cones, all living in the same vector space, such that
(1) if $\sigma$ is a cone in $F$, and $\tau$ is a face of $\sigma$, then $\tau$ is in $F$.
(2) if $\sigma$ and $\sigma'$ are in $F$, then $\sigma \cap \sigma'$ is a face of both $\sigma$ and of $\sigma'$.
Let $\Delta$ be a fan and $\Delta(1)$ be the set of all its edge vectors (one dimensional cones).
Definition 2 : A subset $C\subseteq \Delta(1)$ is a primitive collection if :
$C\nsubseteq \sigma(1)$ for all $\sigma \in \Delta$
For every proper subset $C'$ of $C$ there is a $\sigma\in\Delta$ with $C'\subseteq \sigma(1)$
This is what I understand of the definition 2 -
$C$ is a primitive collection if ALL the edge vectors in $C$ do not span any cone in $\Delta$ but for every proper subset $C'$ of $C$ the edge vectors in $C'$ span a cone in $\Delta$.
Is this understanding correct?
I'm not so sure that I have interpreted condition 2 correctly. My logic was that if $C'\subseteq \sigma(1)$ then either $C'=\sigma(1)$ or $C'\subsetneq\sigma(1)$. If $C'=\sigma(1)$ the $C'$ spans $\sigma$. If $C'\subsetneq \sigma(1)$ the $C'$ spans some face of $\sigma$ which is a cone in $\Delta$ (from definition 1).
But now I am not convinced that $C'$ would always span a face of $\sigma$. Is it true?
Thank you.